{"id":58928,"date":"2025-10-25T09:35:08","date_gmt":"2025-10-25T04:05:08","guid":{"rendered":"https:\/\/therightmentor.com\/?p=58928"},"modified":"2025-10-25T18:14:15","modified_gmt":"2025-10-25T12:44:15","slug":"basics-of-geometry-notes","status":"publish","type":"post","link":"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/","title":{"rendered":"Basics of Geometry | Notes"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 ez-toc-wrap-center counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title ez-toc-toggle\" style=\"cursor:pointer\">Click For Contents <\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#1_Introduction_to_Geometry\" >1. Introduction to Geometry<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#2_Fundamental_Terms\" >2. Fundamental Terms<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#21_Point\" >2.1 Point<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#22_Line\" >2.2 Line<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#23_Line_Segment\" >2.3 Line Segment<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#24_Ray\" >2.4 Ray<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#25_Plane\" >2.5 Plane<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#3_Angle_And_Its_Types\" >3. Angle And Its Types<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#31_Classification\" >3.1 Classification<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#4_Triangle_And_Its_Types\" >4. Triangle And Its Types<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#41_Based_on_Sides\" >4.1 Based on Sides<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#42_Based_on_Angles\" >4.2 Based on Angles<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#5_Quadrilateral_And_Its_Types\" >5. Quadrilateral And Its Types<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#51_Types\" >5.1 Types<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#6_Circles_And_Its_Parts\" >6. Circles\u00a0And Its Parts.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#7_Basic_Geometrical_Constructions\" >7. Basic Geometrical Constructions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#8_Important_Theorems\" >8. Important Theorems<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#9_Tips_Tricks\" >9. Tips &amp; Tricks<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Worksheet_on_Basics_of_Geometry_Math_Olympiad_Preparation\" >Worksheet on Basics of Geometry (Math Olympiad Preparation)<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Section_A_%E2%80%94_Very_Short_Answer_Questions_1_Mark_Each\" >Section A \u2014 Very Short Answer Questions (1 Mark Each)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Section_B_%E2%80%94_Short_Answer_Questions_2_Marks_Each\" >Section B \u2014 Short Answer Questions (2 Marks Each)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Section_C_%E2%80%94_Application_Reasoning_3_Marks_Each\" >Section C \u2014 Application \/ Reasoning (3 Marks Each)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Section_D_%E2%80%94_Higher_Order_Thinking_HOTS_Questions_4%E2%80%935_Marks_Each\" >Section D \u2014 Higher Order Thinking (HOTS) Questions (4\u20135 Marks Each)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Section_E_%E2%80%94_Multiple_Choice_Questions_MCQs\" >Section E \u2014 Multiple Choice Questions (MCQs)<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Worksheet_Hints_Solutions_Answers\" >Worksheet Hints, Solutions &amp; Answers<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Section_A_%E2%80%93_Very_Short_Answer_1_Mark_Each\" >Section A \u2013 Very Short Answer (1 Mark Each)<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q1_Define_a_point_and_a_line\" >Q1. Define a point and a line.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-28\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q2_How_many_endpoints_does_a_line_segment_have\" >Q2. How many endpoints does a line segment have?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-29\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q3_Measure_of_a_straight_angle\" >Q3. Measure of a straight angle?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-30\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q4_Instrument_to_draw_circles\" >Q4. Instrument to draw circles?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-31\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q5_Sum_of_all_angles_around_a_point\" >Q5. Sum of all angles around a point?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-32\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q6_Vertices_of_a_triangle\" >Q6. Vertices of a triangle?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-33\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q7_Sum_of_angles_of_a_quadrilateral\" >Q7. Sum of angles of a quadrilateral?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-34\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q8_Type_of_angle_greater_than_180%C2%B0_but_less_than_360%C2%B0\" >Q8. Type of angle greater than 180\u00b0 but less than 360\u00b0?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-35\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q9_Relationship_between_radius_r_and_diameter_d\" >Q9. Relationship between radius (r) and diameter (d)?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-36\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q10_Example_of_circle_shape\" >Q10. Example of circle shape?<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-37\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Section_B_%E2%80%93_Short_Answer_2_Marks_Each\" >Section B \u2013 Short Answer (2 Marks Each)<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-38\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q11_Draw_and_name_line_segment_ray_line\" >Q11. Draw and name: line segment, ray, line.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-39\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q12_Triangle_with_angles_90%C2%B0_and_45%C2%B0_%E2%86%92_find_third\" >Q12. Triangle with angles 90\u00b0 and 45\u00b0 \u2192 find third.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-40\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q13_Two_angles_sum_130%C2%B0_%E2%86%92_find_supplementary_angles\" >Q13. Two angles sum 130\u00b0 \u2192 find supplementary angles.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-41\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q14_Quadrilateral_angles_80%C2%B0_90%C2%B0_75%C2%B0_%E2%86%92_fourth\" >Q14. Quadrilateral angles 80\u00b0, 90\u00b0, 75\u00b0 \u2192 fourth?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-42\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q15_Circle_radius_7_cm_%E2%86%92_circumference\" >Q15. Circle radius 7 cm \u2192 circumference.<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-43\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Section_C_%E2%80%93_Application_3_Marks_Each\" >Section C \u2013 Application (3 Marks Each)<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-44\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q16_Prove_Adjacent_angles_on_a_straight_line_180%C2%B0\" >Q16. Prove: Adjacent angles on a straight line = 180\u00b0.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-45\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q17_Triangle_sides_6_cm_8_cm_10_cm_%E2%86%92_right_triangle\" >Q17. Triangle sides 6 cm, 8 cm, 10 cm \u2192 right triangle?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-46\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q18_Circle_diameter_14_cm_%E2%86%92_radius_C_A\" >Q18. Circle diameter 14 cm \u2192 radius, C, A.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-47\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q19_Exterior_angle_120%C2%B0_interior_opposite_40%C2%B0_%E2%86%92_other\" >Q19. Exterior angle 120\u00b0, interior opposite 40\u00b0 \u2192 other?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-48\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q20_In_parallelogram_ABCD_%E2%88%A0A_70%C2%B0_%E2%86%92_others\" >Q20. In parallelogram ABCD, \u2220A = 70\u00b0 \u2192 others?<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-49\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Section_D_%E2%80%93_HOTS_4%E2%80%935_Marks_Each\" >Section D \u2013 HOTS (4\u20135 Marks Each)<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-50\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q21_Two_equal_angles_third_96%C2%B0\" >Q21. Two equal angles, third 96\u00b0.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-51\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q22_Sum_of_interior_angles_1440%C2%B0\" >Q22. Sum of interior angles = 1440\u00b0.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-52\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q23_Circle_radius_4_cm_%E2%86%92_mark_points\" >Q23. Circle radius 4 cm \u2192 mark points.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-53\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q24_Quadrilateral_angles_110%C2%B0_95%C2%B0_70%C2%B0_%E2%86%92_fourth\" >Q24. Quadrilateral angles 110\u00b0, 95\u00b0, 70\u00b0 \u2192 fourth?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-54\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Q25_AB8AB_8_cm_bisected_at_MAMMB824AM_MB_frac82_4\" >Q25. AB=8AB = 8 cm bisected at M.AM=MB=82=4AM = MB = \\frac{8}{2} = 4<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-55\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Section_E_%E2%80%93_MCQs\" >Section E \u2013 MCQs<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-56\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Bonus_Challenge_%E2%80%93_Clock_Problem\" >Bonus Challenge \u2013 Clock Problem<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-57\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#1_Write_angles_of_hour_and_minute_hands_in_degrees_at_time_3_t3_t_minutes\" >1) Write angles of hour and minute hands (in degrees) at time 3:t3{:}t minutes<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-58\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#2_Angle_between_the_hands\" >2) Angle between the hands<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-59\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#3_Set_the_angle_equal_to_90%E2%88%9890circ90%E2%88%98_and_solve\" >3) Set the angle equal to 90\u221890^\\circ90\u2218 and solve<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-60\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/#Final_answer_exact_and_approximate\" >Final answer (exact and approximate)<\/a><\/li><\/ul><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 data-start=\"278\" data-end=\"312\"><span class=\"ez-toc-section\" id=\"1_Introduction_to_Geometry\"><\/span><strong data-start=\"281\" data-end=\"312\">1. Introduction to Geometry<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul data-start=\"313\" data-end=\"533\">\n<li data-start=\"313\" data-end=\"443\">\n<p data-start=\"315\" data-end=\"443\">Geometry is the branch of mathematics that deals with <strong data-start=\"369\" data-end=\"442\">shapes, sizes, relative positions of figures, and properties of space<\/strong>.<\/p>\n<\/li>\n<li data-start=\"444\" data-end=\"533\">\n<p data-start=\"446\" data-end=\"533\">The word <strong data-start=\"455\" data-end=\"467\">Geometry<\/strong> comes from Greek words: <code data-start=\"492\" data-end=\"505\">Geo = Earth<\/code> and <code data-start=\"510\" data-end=\"532\">Metron = Measurement<\/code>.<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"535\" data-end=\"538\" \/>\n<h2 data-start=\"540\" data-end=\"567\"><span class=\"ez-toc-section\" id=\"2_Fundamental_Terms\"><\/span><strong data-start=\"543\" data-end=\"567\">2. Fundamental Terms<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 data-start=\"569\" data-end=\"586\"><span class=\"ez-toc-section\" id=\"21_Point\"><\/span><strong data-start=\"573\" data-end=\"586\">2.1 Point<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul data-start=\"587\" data-end=\"753\">\n<li data-start=\"587\" data-end=\"632\">\n<p data-start=\"589\" data-end=\"632\">A <strong data-start=\"591\" data-end=\"600\">point<\/strong> is a precise location in space.<\/p>\n<\/li>\n<li data-start=\"633\" data-end=\"679\">\n<p data-start=\"635\" data-end=\"679\">It has <strong data-start=\"642\" data-end=\"678\">no length, breadth, or thickness<\/strong>.<\/p>\n<\/li>\n<li data-start=\"680\" data-end=\"753\">\n<p data-start=\"682\" data-end=\"753\"><strong data-start=\"682\" data-end=\"695\">Notation:<\/strong> Usually denoted by a capital letter, e.g., <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mo separator=\"true\">,<\/mo><mi>B<\/mi><mo separator=\"true\">,<\/mo><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A, B, C<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">B<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">C<\/span><\/span><\/span><\/span>.<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"755\" data-end=\"771\"><span class=\"ez-toc-section\" id=\"22_Line\"><\/span><strong data-start=\"759\" data-end=\"771\">2.2 Line<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul data-start=\"772\" data-end=\"979\">\n<li data-start=\"772\" data-end=\"858\">\n<p data-start=\"774\" data-end=\"858\">A <strong data-start=\"776\" data-end=\"784\">line<\/strong> is a straight path of points extending <strong data-start=\"824\" data-end=\"857\">infinitely in both directions<\/strong>.<\/p>\n<\/li>\n<li data-start=\"859\" data-end=\"979\">\n<p data-start=\"861\" data-end=\"979\"><strong data-start=\"861\" data-end=\"874\">Notation:<\/strong> Line passing through points <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">B<\/annotation><\/semantics><\/math><\/span><\/span> is written as <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB<\/annotation><\/semantics><\/math><\/span><\/span>\u00a0or <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><mo stretchy=\"true\">\u2194<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\overleftrightarrow{AB}<\/annotation><\/semantics><\/math><\/span><\/span>.<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"981\" data-end=\"1005\"><span class=\"ez-toc-section\" id=\"23_Line_Segment\"><\/span><strong data-start=\"985\" data-end=\"1005\">2.3 Line Segment<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul data-start=\"1006\" data-end=\"1156\">\n<li data-start=\"1006\" data-end=\"1068\">\n<p data-start=\"1008\" data-end=\"1068\">A <strong data-start=\"1010\" data-end=\"1026\">line segment<\/strong> is part of a line with <strong data-start=\"1050\" data-end=\"1067\">two endpoints<\/strong>.<\/p>\n<\/li>\n<li data-start=\"1069\" data-end=\"1156\">\n<p data-start=\"1071\" data-end=\"1156\"><strong data-start=\"1071\" data-end=\"1084\">Notation:<\/strong> Line segment joining points <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">B<\/annotation><\/semantics><\/math><\/span><\/span> is <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><mo stretchy=\"true\">\u203e<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\overline{AB}<\/annotation><\/semantics><\/math><\/span><\/span>.<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"1158\" data-end=\"1173\"><span class=\"ez-toc-section\" id=\"24_Ray\"><\/span><strong data-start=\"1162\" data-end=\"1173\">2.4 Ray<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul data-start=\"1174\" data-end=\"1344\">\n<li data-start=\"1174\" data-end=\"1250\">\n<p data-start=\"1176\" data-end=\"1250\">A <strong data-start=\"1178\" data-end=\"1185\">ray<\/strong> starts at one point and extends <strong data-start=\"1218\" data-end=\"1249\">infinitely in one direction<\/strong>.<\/p>\n<\/li>\n<li data-start=\"1251\" data-end=\"1344\">\n<p data-start=\"1253\" data-end=\"1344\"><strong data-start=\"1253\" data-end=\"1266\">Notation:<\/strong> Ray starting at <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math><\/span><\/span> passing through <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">B<\/annotation><\/semantics><\/math><\/span><\/span> is <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><mo stretchy=\"true\">\u2192<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\overrightarrow{AB}<\/annotation><\/semantics><\/math><\/span><\/span>.<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"1346\" data-end=\"1363\"><span class=\"ez-toc-section\" id=\"25_Plane\"><\/span><strong data-start=\"1350\" data-end=\"1363\">2.5 Plane<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul data-start=\"1364\" data-end=\"1499\">\n<li data-start=\"1364\" data-end=\"1442\">\n<p data-start=\"1366\" data-end=\"1442\">A <strong data-start=\"1368\" data-end=\"1377\">plane<\/strong> is a flat surface that extends <strong data-start=\"1409\" data-end=\"1441\">infinitely in all directions<\/strong>.<\/p>\n<\/li>\n<li data-start=\"1443\" data-end=\"1499\">\n<p data-start=\"1445\" data-end=\"1499\">Represented by a <strong data-start=\"1462\" data-end=\"1486\">quadrilateral figure<\/strong> in diagrams.<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"1501\" data-end=\"1504\" \/>\n<h2 data-start=\"1506\" data-end=\"1531\"><span class=\"ez-toc-section\" id=\"3_Angle_And_Its_Types\"><\/span><strong data-start=\"1509\" data-end=\"1531\">3. Angle And Its <\/strong><strong data-start=\"1509\" data-end=\"1531\">Types<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul data-start=\"1532\" data-end=\"1625\">\n<li data-start=\"1532\" data-end=\"1622\">\n<p data-start=\"1534\" data-end=\"1622\">An <strong data-start=\"1537\" data-end=\"1546\">angle<\/strong> is formed by <strong data-start=\"1560\" data-end=\"1572\">two rays<\/strong> with a <strong data-start=\"1580\" data-end=\"1599\">common endpoint<\/strong> called the <strong data-start=\"1611\" data-end=\"1621\">vertex<\/strong>.<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"1626\" data-end=\"1652\"><span class=\"ez-toc-section\" id=\"31_Classification\"><\/span><strong data-start=\"1630\" data-end=\"1652\">3.1 Classification<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ol data-start=\"1653\" data-end=\"1913\">\n<li data-start=\"1653\" data-end=\"1706\">\n<p data-start=\"1656\" data-end=\"1706\"><strong data-start=\"1656\" data-end=\"1672\">Acute Angle:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>0<\/mn><mo>\u2218<\/mo><\/msup><mo>&lt;<\/mo><mi>\u03b8<\/mi><mo>&lt;<\/mo><msup><mn>90<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">0^\\circ &lt; \\theta &lt; 90^\\circ<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<\/li>\n<li data-start=\"1707\" data-end=\"1750\">\n<p data-start=\"1710\" data-end=\"1750\"><strong data-start=\"1710\" data-end=\"1726\">Right Angle:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><msup><mn>90<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = 90^\\circ<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<\/li>\n<li data-start=\"1751\" data-end=\"1807\">\n<p data-start=\"1754\" data-end=\"1807\"><strong data-start=\"1754\" data-end=\"1771\">Obtuse Angle:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>90<\/mn><mo>\u2218<\/mo><\/msup><mo>&lt;<\/mo><mi>\u03b8<\/mi><mo>&lt;<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">90^\\circ &lt; \\theta &lt; 180^\\circ<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<\/li>\n<li data-start=\"1808\" data-end=\"1855\">\n<p data-start=\"1811\" data-end=\"1855\"><strong data-start=\"1811\" data-end=\"1830\">Straight Angle:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = 180^\\circ<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<\/li>\n<li data-start=\"1856\" data-end=\"1913\">\n<p data-start=\"1859\" data-end=\"1913\"><strong data-start=\"1859\" data-end=\"1876\">Reflex Angle:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><mo>&lt;<\/mo><mi>\u03b8<\/mi><mo>&lt;<\/mo><msup><mn>360<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">180^\\circ &lt; \\theta &lt; 360^\\circ<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<\/li>\n<\/ol>\n<p data-start=\"1915\" data-end=\"1929\"><strong data-start=\"1915\" data-end=\"1927\">Formula:<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mtext>Sum\u00a0of\u00a0angles\u00a0on\u00a0a\u00a0straight\u00a0line<\/mtext><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Sum of angles on a straight line} = 180^\\circ<\/annotation><\/semantics><\/math><\/span><\/span><\/span><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mtext>Sum of angles around a point<\/mtext><mo>=<\/mo><msup><mn>360<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Sum of angles around a point} = 360^\\circ<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<hr data-start=\"2045\" data-end=\"2048\" \/>\n<h2 data-start=\"2050\" data-end=\"2078\"><span class=\"ez-toc-section\" id=\"4_Triangle_And_Its_Types\"><\/span><strong data-start=\"2053\" data-end=\"2078\">4. Triangle <\/strong><strong data-start=\"1509\" data-end=\"1531\">And Its <\/strong><strong data-start=\"1509\" data-end=\"1531\">Types<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul data-start=\"2079\" data-end=\"2129\">\n<li data-start=\"2079\" data-end=\"2129\">\n<p data-start=\"2081\" data-end=\"2129\">A <strong data-start=\"2083\" data-end=\"2095\">triangle<\/strong> has <strong data-start=\"2100\" data-end=\"2111\">3 sides<\/strong> and <strong data-start=\"2116\" data-end=\"2128\">3 angles<\/strong>.<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"2131\" data-end=\"2157\"><span class=\"ez-toc-section\" id=\"41_Based_on_Sides\"><\/span><strong data-start=\"2135\" data-end=\"2157\">4.1 Based on Sides<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ol data-start=\"2158\" data-end=\"2265\">\n<li data-start=\"2158\" data-end=\"2195\">\n<p data-start=\"2161\" data-end=\"2195\"><strong data-start=\"2161\" data-end=\"2177\">Equilateral:<\/strong> All sides equal<\/p>\n<\/li>\n<li data-start=\"2196\" data-end=\"2231\">\n<p data-start=\"2199\" data-end=\"2231\"><strong data-start=\"2199\" data-end=\"2213\">Isosceles:<\/strong> Two sides equal<\/p>\n<\/li>\n<li data-start=\"2232\" data-end=\"2265\">\n<p data-start=\"2235\" data-end=\"2265\"><strong data-start=\"2235\" data-end=\"2247\">Scalene:<\/strong> All sides unequal<\/p>\n<\/li>\n<\/ol>\n<h3 data-start=\"2267\" data-end=\"2294\"><span class=\"ez-toc-section\" id=\"42_Based_on_Angles\"><\/span><strong data-start=\"2271\" data-end=\"2294\">4.2 Based on Angles<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ol data-start=\"2295\" data-end=\"2444\">\n<li data-start=\"2295\" data-end=\"2345\">\n<p data-start=\"2298\" data-end=\"2345\"><strong data-start=\"2298\" data-end=\"2315\">Acute-angled:<\/strong> All angles &lt; <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>90<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">90^\\circ<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<\/li>\n<li data-start=\"2346\" data-end=\"2395\">\n<p data-start=\"2349\" data-end=\"2395\"><strong data-start=\"2349\" data-end=\"2366\">Right-angled:<\/strong> One angle = <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>90<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">90^\\circ<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<\/li>\n<li data-start=\"2396\" data-end=\"2444\">\n<p data-start=\"2399\" data-end=\"2444\"><strong data-start=\"2399\" data-end=\"2417\">Obtuse-angled:<\/strong> One angle &gt; <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>90<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">90^\\circ<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<\/li>\n<\/ol>\n<p data-start=\"2446\" data-end=\"2494\"><strong data-start=\"2446\" data-end=\"2492\">Important Property (Triangle Sum Theorem):<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mtext>Sum\u00a0of\u00a0angles\u00a0in\u00a0a\u00a0triangle<\/mtext><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Sum of angles in a triangle} = 180^\\circ<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<h2 data-start=\"2554\" data-end=\"2578\"><span class=\"ez-toc-section\" id=\"5_Quadrilateral_And_Its_Types\"><\/span><strong data-start=\"2557\" data-end=\"2578\">5. Quadrilateral <\/strong><strong data-start=\"1509\" data-end=\"1531\">And Its <\/strong><strong data-start=\"1509\" data-end=\"1531\">Types<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul data-start=\"2579\" data-end=\"2634\">\n<li data-start=\"2579\" data-end=\"2634\">\n<p data-start=\"2581\" data-end=\"2634\">A <strong data-start=\"2583\" data-end=\"2600\">quadrilateral<\/strong> has <strong data-start=\"2605\" data-end=\"2616\">4 sides<\/strong> and <strong data-start=\"2621\" data-end=\"2633\">4 angles<\/strong>.<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"2636\" data-end=\"2653\"><span class=\"ez-toc-section\" id=\"51_Types\"><\/span><strong data-start=\"2640\" data-end=\"2653\">5.1 Types<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ol data-start=\"2654\" data-end=\"2952\">\n<li data-start=\"2654\" data-end=\"2713\">\n<p data-start=\"2657\" data-end=\"2713\"><strong data-start=\"2657\" data-end=\"2668\">Square:<\/strong> All sides equal, all angles <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>90<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">90^\\circ<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<\/li>\n<li data-start=\"2714\" data-end=\"2781\">\n<p data-start=\"2717\" data-end=\"2781\"><strong data-start=\"2717\" data-end=\"2731\">Rectangle:<\/strong> Opposite sides equal, all angles <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>90<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">90^\\circ<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<\/li>\n<li data-start=\"2782\" data-end=\"2839\">\n<p data-start=\"2785\" data-end=\"2839\"><strong data-start=\"2785\" data-end=\"2803\">Parallelogram:<\/strong> Opposite sides parallel and equal<\/p>\n<\/li>\n<li data-start=\"2840\" data-end=\"2896\">\n<p data-start=\"2843\" data-end=\"2896\"><strong data-start=\"2843\" data-end=\"2855\">Rhombus:<\/strong> All sides equal, opposite angles equal<\/p>\n<\/li>\n<li data-start=\"2897\" data-end=\"2952\">\n<p data-start=\"2900\" data-end=\"2952\"><strong data-start=\"2900\" data-end=\"2914\">Trapezium:<\/strong> One pair of opposite sides parallel<\/p>\n<\/li>\n<\/ol>\n<p data-start=\"2954\" data-end=\"3003\"><strong data-start=\"2954\" data-end=\"2967\">Property:<\/strong> Sum of angles in a quadrilateral:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mtext>Sum\u00a0of\u00a0angles<\/mtext><mo>=<\/mo><msup><mn>360<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Sum of angles} = 360^\\circ<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<h2 data-start=\"3049\" data-end=\"3066\"><span class=\"ez-toc-section\" id=\"6_Circles_And_Its_Parts\"><\/span><strong data-start=\"3052\" data-end=\"3066\">6. Circles\u00a0<\/strong><strong data-start=\"1509\" data-end=\"1531\">And Its <\/strong><strong data-start=\"1509\" data-end=\"1531\">Parts.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul data-start=\"3067\" data-end=\"3358\">\n<li data-start=\"3067\" data-end=\"3158\">\n<p data-start=\"3069\" data-end=\"3158\">A <strong data-start=\"3071\" data-end=\"3081\">circle<\/strong> is a set of points <strong data-start=\"3101\" data-end=\"3116\">equidistant<\/strong> from a fixed point called the <strong data-start=\"3147\" data-end=\"3157\">centre<\/strong>.<\/p>\n<\/li>\n<li data-start=\"3159\" data-end=\"3226\">\n<p data-start=\"3161\" data-end=\"3226\"><strong data-start=\"3161\" data-end=\"3176\">Radius (r):<\/strong> Distance from centre to any point on the circle<\/p>\n<\/li>\n<li data-start=\"3227\" data-end=\"3279\">\n<p data-start=\"3229\" data-end=\"3279\"><strong data-start=\"3229\" data-end=\"3246\">Diameter (d):<\/strong> Twice the radius, <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mo>=<\/mo><mn>2<\/mn><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">d = 2r<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<\/li>\n<li data-start=\"3280\" data-end=\"3324\">\n<p data-start=\"3282\" data-end=\"3324\"><strong data-start=\"3282\" data-end=\"3304\">Circumference (C):<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mo>=<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">C = 2 \\pi r<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<\/li>\n<li data-start=\"3325\" data-end=\"3358\">\n<p data-start=\"3327\" data-end=\"3358\"><strong data-start=\"3327\" data-end=\"3340\">Area (A):<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mi>\u03c0<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">A = \\pi r^2<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"3360\" data-end=\"3363\" \/>\n<h2 data-start=\"3365\" data-end=\"3406\"><span class=\"ez-toc-section\" id=\"7_Basic_Geometrical_Constructions\"><\/span><strong data-start=\"3368\" data-end=\"3406\">7. Basic Geometrical Constructions<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ol data-start=\"3407\" data-end=\"3641\">\n<li data-start=\"3407\" data-end=\"3457\">\n<p data-start=\"3410\" data-end=\"3457\"><strong data-start=\"3410\" data-end=\"3455\">Constructing a bisector of a line segment<\/strong><\/p>\n<\/li>\n<li data-start=\"3458\" data-end=\"3497\">\n<p data-start=\"3461\" data-end=\"3497\"><strong data-start=\"3461\" data-end=\"3495\">Constructing an angle bisector<\/strong><\/p>\n<\/li>\n<li data-start=\"3498\" data-end=\"3575\">\n<p data-start=\"3501\" data-end=\"3575\"><strong data-start=\"3501\" data-end=\"3573\">Constructing perpendiculars from a point on a line or outside a line<\/strong><\/p>\n<\/li>\n<li data-start=\"3576\" data-end=\"3641\">\n<p data-start=\"3579\" data-end=\"3641\"><strong data-start=\"3579\" data-end=\"3639\">Constructing triangles using SSS, SAS, ASA, RHS criteria<\/strong><\/p>\n<\/li>\n<\/ol>\n<hr data-start=\"3643\" data-end=\"3646\" \/>\n<h2 data-start=\"3648\" data-end=\"3676\"><span class=\"ez-toc-section\" id=\"8_Important_Theorems\"><\/span><strong data-start=\"3651\" data-end=\"3676\">8. Important Theorems<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ol data-start=\"3677\" data-end=\"3705\">\n<li data-start=\"3677\" data-end=\"3705\">\n<p data-start=\"3680\" data-end=\"3705\"><strong data-start=\"3680\" data-end=\"3703\">Pythagoras Theorem:<\/strong><\/p>\n<\/li>\n<\/ol>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mtext>In\u00a0a\u00a0right\u00a0triangle:\u00a0<\/mtext><mi>A<\/mi><msup><mi>B<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>B<\/mi><msup><mi>C<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mi>A<\/mi><msup><mi>C<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\text{In a right triangle: } AB^2 + BC^2 = AC^2<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<ol start=\"2\" data-start=\"3760\" data-end=\"3790\">\n<li data-start=\"3760\" data-end=\"3790\">\n<p data-start=\"3763\" data-end=\"3790\"><strong data-start=\"3763\" data-end=\"3788\">Triangle Sum Theorem:<\/strong><\/p>\n<\/li>\n<\/ol>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u2220<\/mi><mi>A<\/mi><mo>+<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>B<\/mi><mo>+<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>C<\/mi><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\angle A + \\angle B + \\angle C = 180^\\circ<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<ol start=\"3\" data-start=\"3840\" data-end=\"3872\">\n<li data-start=\"3840\" data-end=\"3872\">\n<p data-start=\"3843\" data-end=\"3872\"><strong data-start=\"3843\" data-end=\"3870\">Exterior Angle Theorem:<\/strong><\/p>\n<\/li>\n<\/ol>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mtext>Exterior\u00a0angle\u00a0of\u00a0a\u00a0triangle<\/mtext><mo>=<\/mo><mtext>Sum\u00a0of\u00a0the\u00a0two\u00a0opposite\u00a0interior\u00a0angles<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Exterior angle of a triangle} = \\text{Sum of the two opposite interior angles}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<h2 data-start=\"3970\" data-end=\"3993\"><span class=\"ez-toc-section\" id=\"9_Tips_Tricks\"><\/span><strong data-start=\"3973\" data-end=\"3993\">9. Tips &amp; Tricks<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul data-start=\"3994\" data-end=\"4263\">\n<li data-start=\"3994\" data-end=\"4038\">\n<p data-start=\"3996\" data-end=\"4038\">Always label points clearly in diagrams.<\/p>\n<\/li>\n<li data-start=\"4039\" data-end=\"4095\">\n<p data-start=\"4041\" data-end=\"4095\">Use a <strong data-start=\"4047\" data-end=\"4061\">protractor<\/strong> for accurate angle measurement.<\/p>\n<\/li>\n<li data-start=\"4096\" data-end=\"4177\">\n<p data-start=\"4098\" data-end=\"4177\">Remember the sum of angles for <strong data-start=\"4129\" data-end=\"4148\">triangle = 180\u00b0<\/strong>, <strong data-start=\"4150\" data-end=\"4174\">quadrilateral = 360\u00b0<\/strong>.<\/p>\n<\/li>\n<li data-start=\"4178\" data-end=\"4263\">\n<p data-start=\"4180\" data-end=\"4263\">Practice constructing triangles using different combinations of sides and angles.<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"4265\" data-end=\"4268\" \/>\n<p data-start=\"4270\" data-end=\"4297\"><strong data-start=\"4270\" data-end=\"4295\">Diagram Placeholders:<\/strong><\/p>\n<ul data-start=\"4298\" data-end=\"4431\">\n<li data-start=\"4298\" data-end=\"4334\">\n<p data-start=\"4300\" data-end=\"4334\">[Point, Line, Line Segment, Ray]\n<\/li>\n<li data-start=\"4335\" data-end=\"4369\">\n<p data-start=\"4337\" data-end=\"4369\">[Triangle with labeled angles]\n<\/li>\n<li data-start=\"4370\" data-end=\"4395\">\n<p data-start=\"4372\" data-end=\"4395\">[Quadrilateral types]\n<\/li>\n<li data-start=\"4396\" data-end=\"4431\">\n<p data-start=\"4398\" data-end=\"4431\">[Circle with radius and diameter]\n<\/li>\n<\/ul>\n<h1 data-start=\"294\" data-end=\"364\"><span class=\"ez-toc-section\" id=\"Worksheet_on_Basics_of_Geometry_Math_Olympiad_Preparation\"><\/span><strong data-start=\"299\" data-end=\"362\">Worksheet on Basics of Geometry (Math Olympiad Preparation)<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p data-start=\"365\" data-end=\"495\"><strong data-start=\"365\" data-end=\"375\">Topic:<\/strong> Basics of Geometry<br data-start=\"394\" data-end=\"397\" \/><strong data-start=\"397\" data-end=\"407\">Class:<\/strong> 6\u20139 (CBSE &amp; Olympiad Level)<br data-start=\"435\" data-end=\"438\" \/><strong data-start=\"438\" data-end=\"448\">Marks:<\/strong> Practice Worksheet<br data-start=\"467\" data-end=\"470\" \/><strong data-start=\"470\" data-end=\"479\">Time:<\/strong> 45\u201360 minutes<\/p>\n<hr data-start=\"497\" data-end=\"500\" \/>\n<h2 data-start=\"502\" data-end=\"564\"><span class=\"ez-toc-section\" id=\"Section_A_%E2%80%94_Very_Short_Answer_Questions_1_Mark_Each\"><\/span><strong data-start=\"505\" data-end=\"562\">Section A \u2014 Very Short Answer Questions (1 Mark Each)<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"565\" data-end=\"599\"><em data-start=\"565\" data-end=\"597\">(Concept Recall &amp; Definitions)<\/em><\/p>\n<ol data-start=\"601\" data-end=\"1172\">\n<li data-start=\"601\" data-end=\"652\">\n<p data-start=\"604\" data-end=\"652\">Define a <strong data-start=\"613\" data-end=\"622\">point<\/strong> and a <strong data-start=\"629\" data-end=\"637\">line<\/strong> in geometry.<\/p>\n<\/li>\n<li data-start=\"653\" data-end=\"706\">\n<p data-start=\"656\" data-end=\"706\">How many <strong data-start=\"665\" data-end=\"678\">endpoints<\/strong> does a line segment have?<\/p>\n<\/li>\n<li data-start=\"707\" data-end=\"756\">\n<p data-start=\"710\" data-end=\"756\">What is the <strong data-start=\"722\" data-end=\"753\">measure of a straight angle<\/strong>?<\/p>\n<\/li>\n<li data-start=\"757\" data-end=\"807\">\n<p data-start=\"760\" data-end=\"807\">Name the <strong data-start=\"769\" data-end=\"783\">instrument<\/strong> used to draw circles.<\/p>\n<\/li>\n<li data-start=\"808\" data-end=\"860\">\n<p data-start=\"811\" data-end=\"860\">Write the <strong data-start=\"821\" data-end=\"857\">sum of all angles around a point<\/strong>.<\/p>\n<\/li>\n<li data-start=\"861\" data-end=\"909\">\n<p data-start=\"864\" data-end=\"909\">How many <strong data-start=\"873\" data-end=\"885\">vertices<\/strong> does a triangle have?<\/p>\n<\/li>\n<li data-start=\"910\" data-end=\"968\">\n<p data-start=\"913\" data-end=\"968\">What is the <strong data-start=\"925\" data-end=\"946\">sum of the angles<\/strong> of a quadrilateral?<\/p>\n<\/li>\n<li data-start=\"969\" data-end=\"1038\">\n<p data-start=\"972\" data-end=\"1038\">Which <strong data-start=\"978\" data-end=\"995\">type of angle<\/strong> is greater than 180\u00b0 but less than 360\u00b0?<\/p>\n<\/li>\n<li data-start=\"1039\" data-end=\"1099\">\n<p data-start=\"1042\" data-end=\"1099\">Write the <strong data-start=\"1052\" data-end=\"1096\">relationship between radius and diameter<\/strong>.<\/p>\n<\/li>\n<li data-start=\"1100\" data-end=\"1172\">\n<p data-start=\"1104\" data-end=\"1172\">Give one <strong data-start=\"1113\" data-end=\"1146\">example of a real-life object<\/strong> in the shape of a circle.<\/p>\n<\/li>\n<\/ol>\n<hr data-start=\"1174\" data-end=\"1177\" \/>\n<h2 data-start=\"1179\" data-end=\"1237\"><span class=\"ez-toc-section\" id=\"Section_B_%E2%80%94_Short_Answer_Questions_2_Marks_Each\"><\/span><strong data-start=\"1182\" data-end=\"1235\">Section B \u2014 Short Answer Questions (2 Marks Each)<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"1238\" data-end=\"1271\"><em data-start=\"1238\" data-end=\"1269\">(Understanding &amp; Application)<\/em><\/p>\n<ol start=\"11\" data-start=\"1273\" data-end=\"1816\">\n<li data-start=\"1273\" data-end=\"1350\">\n<p data-start=\"1277\" data-end=\"1350\">Draw and name the following:<br data-start=\"1305\" data-end=\"1308\" \/>\u2003(a) Line segment<br data-start=\"1325\" data-end=\"1328\" \/>\u2003(b) Ray<br data-start=\"1336\" data-end=\"1339\" \/>\u2003(c) Line<\/p>\n<\/li>\n<li data-start=\"1352\" data-end=\"1455\">\n<p data-start=\"1356\" data-end=\"1455\">If one angle of a triangle is <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>90<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">90^\\circ<\/annotation><\/semantics><\/math><\/span><\/span> and another is <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>45<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">45^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>, find the third angle.<\/p>\n<\/li>\n<li data-start=\"1457\" data-end=\"1556\">\n<p data-start=\"1461\" data-end=\"1556\">The sum of two angles is <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>130<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">130^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>. Find the measure of their <strong data-start=\"1529\" data-end=\"1546\">supplementary<\/strong> angles.<\/p>\n<\/li>\n<li data-start=\"1558\" data-end=\"1659\">\n<p data-start=\"1562\" data-end=\"1659\">In a quadrilateral, three angles are <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>80<\/mn><mo>\u2218<\/mo><\/msup><mo separator=\"true\">,<\/mo><msup><mn>90<\/mn><mo>\u2218<\/mo><\/msup><mo separator=\"true\">,<\/mo><msup><mn>75<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">80^\\circ, 90^\\circ, 75^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>. Find the fourth angle.<\/p>\n<\/li>\n<li data-start=\"1661\" data-end=\"1816\">\n<p data-start=\"1665\" data-end=\"1816\">A circle has a <strong data-start=\"1680\" data-end=\"1698\">radius of 7 cm<\/strong>. Find its <strong data-start=\"1709\" data-end=\"1726\">circumference<\/strong> using <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c0<\/mi><mo>=<\/mo><mfrac><mn>22<\/mn><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\pi = \\frac{22}{7}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<br data-start=\"1758\" data-end=\"1761\" \/>\u2003<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mo>=<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mi>r<\/mi><mo>=<\/mo><mn>2<\/mn><mo>\u00d7<\/mo><mfrac><mn>22<\/mn><mn>7<\/mn><\/mfrac><mo>\u00d7<\/mo><mn>7<\/mn><mo>=<\/mo><mo stretchy=\"false\">?<\/mo><\/mrow><\/semantics><\/math><\/span><\/span><\/p>\n<\/li>\n<\/ol>\n<hr data-start=\"1818\" data-end=\"1821\" \/>\n<h2 data-start=\"1823\" data-end=\"1882\"><span class=\"ez-toc-section\" id=\"Section_C_%E2%80%94_Application_Reasoning_3_Marks_Each\"><\/span><strong data-start=\"1826\" data-end=\"1880\">Section C \u2014 Application \/ Reasoning (3 Marks Each)<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ol start=\"16\" data-start=\"1884\" data-end=\"2627\">\n<li data-start=\"1884\" data-end=\"2016\">\n<p data-start=\"1888\" data-end=\"2016\">The sum of two adjacent angles on a straight line is always <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">180^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>. Prove this statement using a <strong data-start=\"1997\" data-end=\"2013\">neat diagram<\/strong>.<\/p>\n<\/li>\n<li data-start=\"2018\" data-end=\"2178\">\n<p data-start=\"2022\" data-end=\"2178\">A triangle has sides 6 cm, 8 cm, and 10 cm. Verify whether it is a <strong data-start=\"2092\" data-end=\"2117\">right-angled triangle<\/strong> using the <strong data-start=\"2128\" data-end=\"2150\">Pythagoras Theorem<\/strong>.<br data-start=\"2151\" data-end=\"2154\" \/>\u2003<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>c<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">a^2 + b^2 = c^2<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<\/li>\n<li data-start=\"2180\" data-end=\"2308\">\n<p data-start=\"2184\" data-end=\"2308\">In a circle with <strong data-start=\"2201\" data-end=\"2219\">diameter 14 cm<\/strong>, find:<br data-start=\"2226\" data-end=\"2229\" \/>\u2003(a) Radius<br data-start=\"2240\" data-end=\"2243\" \/>\u2003(b) Circumference<br data-start=\"2261\" data-end=\"2264\" \/>\u2003(c) Area<br data-start=\"2273\" data-end=\"2276\" \/>\u2003Use <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c0<\/mi><mo>=<\/mo><mfrac><mn>22<\/mn><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\pi = \\frac{22}{7}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">\u03c0<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">7<\/span><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">22<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/p>\n<\/li>\n<li data-start=\"2310\" data-end=\"2469\">\n<p data-start=\"2314\" data-end=\"2469\">The exterior angle of a triangle is <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>120<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">120^\\circ<\/annotation><\/semantics><\/math><\/span><\/span> and one of the interior opposite angles is <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>40<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">40^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>.<br data-start=\"2424\" data-end=\"2427\" \/>\u2003Find the other interior opposite angle.<\/p>\n<\/li>\n<li data-start=\"2471\" data-end=\"2627\">\n<p data-start=\"2475\" data-end=\"2627\">In parallelogram <strong data-start=\"2492\" data-end=\"2500\">ABCD<\/strong>, <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u2220<\/mi><mi>A<\/mi><mo>=<\/mo><msup><mn>70<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\angle A = 70^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>. Find all other angles.<br data-start=\"2551\" data-end=\"2554\" \/>\u2003(Hint: Opposite angles are equal and adjacent angles are supplementary.)<\/p>\n<\/li>\n<\/ol>\n<hr data-start=\"2629\" data-end=\"2632\" \/>\n<h2 data-start=\"2634\" data-end=\"2710\"><span class=\"ez-toc-section\" id=\"Section_D_%E2%80%94_Higher_Order_Thinking_HOTS_Questions_4%E2%80%935_Marks_Each\"><\/span><strong data-start=\"2637\" data-end=\"2708\">Section D \u2014 Higher Order Thinking (HOTS) Questions (4\u20135 Marks Each)<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ol start=\"21\" data-start=\"2712\" data-end=\"3381\">\n<li data-start=\"2712\" data-end=\"2844\">\n<p data-start=\"2716\" data-end=\"2844\">A triangle has two equal angles and the third angle is <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>96<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">96^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>.<br data-start=\"2786\" data-end=\"2789\" \/>\u2003Find each of the equal angles and name the triangle.<\/p>\n<\/li>\n<li data-start=\"2846\" data-end=\"2961\">\n<p data-start=\"2850\" data-end=\"2961\">The sum of the interior angles of an <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span>-sided polygon is <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>1440<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">1440^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>.<br data-start=\"2929\" data-end=\"2932\" \/>\u2003Find the value of <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math><\/span><\/span>.<\/p>\n<\/li>\n<li data-start=\"2963\" data-end=\"3087\">\n<p data-start=\"2967\" data-end=\"3087\">Draw a <strong data-start=\"2974\" data-end=\"2999\">circle of radius 4 cm<\/strong>, mark points:<br data-start=\"3013\" data-end=\"3016\" \/>\u2003(a) Inside the circle<br data-start=\"3038\" data-end=\"3041\" \/>\u2003(b) On the circle<br data-start=\"3059\" data-end=\"3062\" \/>\u2003(c) Outside the circle<\/p>\n<\/li>\n<li data-start=\"3089\" data-end=\"3231\">\n<p data-start=\"3093\" data-end=\"3231\">A quadrilateral has three angles measuring <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>110<\/mn><mo>\u2218<\/mo><\/msup><mo separator=\"true\">,<\/mo><msup><mn>95<\/mn><mo>\u2218<\/mo><\/msup><mo separator=\"true\">,<\/mo><msup><mn>70<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">110^\\circ, 95^\\circ, 70^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>. Find the fourth angle and classify the quadrilateral.<\/p>\n<\/li>\n<li data-start=\"3233\" data-end=\"3381\">\n<p data-start=\"3237\" data-end=\"3381\">A line segment <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mo>=<\/mo><mn>8<\/mn><mtext>\u00a0cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">AB = 8 \\text{ cm}<\/annotation><\/semantics><\/math><\/span><\/span> is bisected at <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>M<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">M<\/annotation><\/semantics><\/math><\/span><\/span>.<br data-start=\"3299\" data-end=\"3302\" \/>\u2003Find <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>M<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AM<\/annotation><\/semantics><\/math><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>M<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">MB<\/annotation><\/semantics><\/math><\/span><\/span>, and justify your answer using geometric reasoning.<\/p>\n<\/li>\n<\/ol>\n<hr data-start=\"3383\" data-end=\"3386\" \/>\n<h2 data-start=\"3388\" data-end=\"3441\"><span class=\"ez-toc-section\" id=\"Section_E_%E2%80%94_Multiple_Choice_Questions_MCQs\"><\/span><strong data-start=\"3391\" data-end=\"3439\">Section E \u2014 Multiple Choice Questions (MCQs)<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"3442\" data-end=\"3466\"><em data-start=\"3442\" data-end=\"3464\">(For Quick Revision)<\/em><\/p>\n<ol start=\"26\" data-start=\"3468\" data-end=\"4116\">\n<li data-start=\"3468\" data-end=\"3548\">\n<p data-start=\"3472\" data-end=\"3548\">The total number of right angles in a rectangle is:<br data-start=\"3523\" data-end=\"3526\" \/>\u2003A. 1\u2003B. 2\u2003C. 3\u2003D. 4<\/p>\n<\/li>\n<li data-start=\"3550\" data-end=\"3646\">\n<p data-start=\"3554\" data-end=\"3646\">The angle formed by two perpendicular lines is:<br data-start=\"3601\" data-end=\"3604\" \/>\u2003A. Acute\u2003B. Right\u2003C. Obtuse\u2003D. Straight<\/p>\n<\/li>\n<li data-start=\"3648\" data-end=\"3771\">\n<p data-start=\"3652\" data-end=\"3771\">The sum of all angles in a triangle is:\u00a0<br data-start=\"3691\" data-end=\"3694\" \/>\u2003A. <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>90<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">90^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>\u2003B. <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">180^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>\u2003C. <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>270<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">270^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>\u2003D. <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>360<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">360^\\circ<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<\/li>\n<li data-start=\"3773\" data-end=\"3920\">\n<p data-start=\"3777\" data-end=\"3920\">A line segment joining the <strong data-start=\"3804\" data-end=\"3826\">centre of a circle<\/strong> to a point on its <strong data-start=\"3845\" data-end=\"3862\">circumference<\/strong> is called:<br data-start=\"3873\" data-end=\"3876\" \/>\u2003A. Chord\u2003B. Diameter\u2003C. Radius\u2003D. Tangent<\/p>\n<\/li>\n<li data-start=\"3922\" data-end=\"4116\">\n<p data-start=\"3926\" data-end=\"4116\">Which of the following statements is <strong data-start=\"3963\" data-end=\"3972\">false<\/strong>?<br data-start=\"3973\" data-end=\"3976\" \/>\u2003A. Every square is a rectangle.<br data-start=\"4008\" data-end=\"4011\" \/>\u2003B. Every rectangle is a square.<br data-start=\"4043\" data-end=\"4046\" \/>\u2003C. Every square is a rhombus.<br data-start=\"4076\" data-end=\"4079\" \/>\u2003D. Every rhombus is a parallelogram.<\/p>\n<\/li>\n<\/ol>\n<h1 data-start=\"389\" data-end=\"463\"><span class=\"ez-toc-section\" id=\"Worksheet_Hints_Solutions_Answers\"><\/span><strong data-start=\"394\" data-end=\"463\">Worksheet Hints, Solutions &amp; Answers<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h1>\n<hr data-start=\"465\" data-end=\"468\" \/>\n<h2 data-start=\"470\" data-end=\"520\"><span class=\"ez-toc-section\" id=\"Section_A_%E2%80%93_Very_Short_Answer_1_Mark_Each\"><\/span><strong data-start=\"473\" data-end=\"520\">Section A \u2013 Very Short Answer (1 Mark Each)<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 data-start=\"522\" data-end=\"562\"><span class=\"ez-toc-section\" id=\"Q1_Define_a_point_and_a_line\"><\/span><strong data-start=\"526\" data-end=\"533\">Q1.<\/strong> Define a point and a line.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"563\" data-end=\"630\"><strong data-start=\"563\" data-end=\"572\">Hint:<\/strong> Recall definitions from basic geometry.<br data-start=\"612\" data-end=\"615\" \/><strong data-start=\"615\" data-end=\"628\">Solution:<\/strong><\/p>\n<ul data-start=\"631\" data-end=\"822\">\n<li data-start=\"631\" data-end=\"672\">\n<p data-start=\"633\" data-end=\"672\">A <strong data-start=\"635\" data-end=\"644\">point<\/strong> has position but no size.<\/p>\n<\/li>\n<li data-start=\"673\" data-end=\"822\">\n<p data-start=\"675\" data-end=\"822\">A <strong data-start=\"677\" data-end=\"685\">line<\/strong> extends endlessly in both directions, made of infinite points.<br data-start=\"748\" data-end=\"751\" \/><strong data-start=\"751\" data-end=\"762\">Answer:<\/strong> Point \u2013 no dimensions; Line \u2013 extends infinitely both ways.<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"824\" data-end=\"827\" \/>\n<h3 data-start=\"829\" data-end=\"887\"><span class=\"ez-toc-section\" id=\"Q2_How_many_endpoints_does_a_line_segment_have\"><\/span><strong data-start=\"833\" data-end=\"840\">Q2.<\/strong> How many endpoints does a line segment have?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"888\" data-end=\"1011\"><strong data-start=\"888\" data-end=\"897\">Hint:<\/strong> Think of a line with fixed ends.<br data-start=\"930\" data-end=\"933\" \/><strong data-start=\"933\" data-end=\"946\">Solution:<\/strong> A line segment has <strong data-start=\"966\" data-end=\"973\">two<\/strong> endpoints.<br data-start=\"984\" data-end=\"987\" \/><strong data-start=\"987\" data-end=\"998\">Answer:<\/strong> 2 endpoints.<\/p>\n<hr data-start=\"1013\" data-end=\"1016\" \/>\n<h3 data-start=\"1018\" data-end=\"1060\"><span class=\"ez-toc-section\" id=\"Q3_Measure_of_a_straight_angle\"><\/span><strong data-start=\"1022\" data-end=\"1029\">Q3.<\/strong> Measure of a straight angle?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"1061\" data-end=\"1157\"><strong data-start=\"1061\" data-end=\"1070\">Hint:<\/strong> It lies on a straight line.<br data-start=\"1098\" data-end=\"1101\" \/><strong data-start=\"1101\" data-end=\"1114\">Solution:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">180^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>.<br data-start=\"1129\" data-end=\"1132\" \/><strong data-start=\"1132\" data-end=\"1143\">Answer:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">180^\\circ<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<hr data-start=\"1159\" data-end=\"1162\" \/>\n<h3 data-start=\"1164\" data-end=\"1205\"><span class=\"ez-toc-section\" id=\"Q4_Instrument_to_draw_circles\"><\/span><strong data-start=\"1168\" data-end=\"1175\">Q4.<\/strong> Instrument to draw circles?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"1206\" data-end=\"1264\"><strong data-start=\"1206\" data-end=\"1215\">Hint:<\/strong> Used with pencil and pin.<br data-start=\"1241\" data-end=\"1244\" \/><strong data-start=\"1244\" data-end=\"1255\">Answer:<\/strong> Compass.<\/p>\n<hr data-start=\"1266\" data-end=\"1269\" \/>\n<h3 data-start=\"1271\" data-end=\"1318\"><span class=\"ez-toc-section\" id=\"Q5_Sum_of_all_angles_around_a_point\"><\/span><strong data-start=\"1275\" data-end=\"1282\">Q5.<\/strong> Sum of all angles around a point?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"1319\" data-end=\"1334\"><strong data-start=\"1319\" data-end=\"1332\">Solution:<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mtext>Sum\u00a0of\u00a0all\u00a0angles\u00a0around\u00a0a\u00a0point<\/mtext><mo>=<\/mo><msup><mn>360<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Sum of all angles around a point} = 360^\\circ<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"1395\" data-end=\"1420\"><strong data-start=\"1395\" data-end=\"1406\">Answer:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>360<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">360^\\circ<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<hr data-start=\"1422\" data-end=\"1425\" \/>\n<h3 data-start=\"1427\" data-end=\"1464\"><span class=\"ez-toc-section\" id=\"Q6_Vertices_of_a_triangle\"><\/span><strong data-start=\"1431\" data-end=\"1438\">Q6.<\/strong> Vertices of a triangle?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"1465\" data-end=\"1488\"><strong data-start=\"1465\" data-end=\"1476\">Answer:<\/strong> 3 vertices.<\/p>\n<hr data-start=\"1490\" data-end=\"1493\" \/>\n<h3 data-start=\"1495\" data-end=\"1542\"><span class=\"ez-toc-section\" id=\"Q7_Sum_of_angles_of_a_quadrilateral\"><\/span><strong data-start=\"1499\" data-end=\"1506\">Q7.<\/strong> Sum of angles of a quadrilateral?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"1543\" data-end=\"1558\"><strong data-start=\"1543\" data-end=\"1556\">Solution:<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mtext>Sum<\/mtext><mo>=<\/mo><msup><mn>360<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Sum} = 360^\\circ<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"1590\" data-end=\"1615\"><strong data-start=\"1590\" data-end=\"1601\">Answer:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>360<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">360^\\circ<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<hr data-start=\"1617\" data-end=\"1620\" \/>\n<h3 data-start=\"1622\" data-end=\"1687\"><span class=\"ez-toc-section\" id=\"Q8_Type_of_angle_greater_than_180%C2%B0_but_less_than_360%C2%B0\"><\/span><strong data-start=\"1626\" data-end=\"1633\">Q8.<\/strong> Type of angle greater than 180\u00b0 but less than 360\u00b0?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"1688\" data-end=\"1713\"><strong data-start=\"1688\" data-end=\"1699\">Answer:<\/strong> Reflex angle.<\/p>\n<hr data-start=\"1715\" data-end=\"1718\" \/>\n<h3 data-start=\"1720\" data-end=\"1783\"><span class=\"ez-toc-section\" id=\"Q9_Relationship_between_radius_r_and_diameter_d\"><\/span><strong data-start=\"1724\" data-end=\"1731\">Q9.<\/strong> Relationship between radius (r) and diameter (d)?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mo>=<\/mo><mn>2<\/mn><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">d = 2r<\/annotation><\/semantics><\/math><\/span><\/span><\/span><strong data-start=\"1799\" data-end=\"1810\">Answer:<\/strong> Diameter = 2 \u00d7 Radius.<\/p>\n<h3 data-start=\"1840\" data-end=\"1879\"><span class=\"ez-toc-section\" id=\"Q10_Example_of_circle_shape\"><\/span><strong data-start=\"1844\" data-end=\"1852\">Q10.<\/strong> Example of circle shape?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"1880\" data-end=\"1911\"><strong data-start=\"1880\" data-end=\"1891\">Answer:<\/strong> Clock, coin, wheel.<\/p>\n<hr data-start=\"1913\" data-end=\"1916\" \/>\n<h2 data-start=\"1918\" data-end=\"1964\"><span class=\"ez-toc-section\" id=\"Section_B_%E2%80%93_Short_Answer_2_Marks_Each\"><\/span><strong data-start=\"1921\" data-end=\"1964\">Section B \u2013 Short Answer (2 Marks Each)<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 data-start=\"1966\" data-end=\"2020\"><span class=\"ez-toc-section\" id=\"Q11_Draw_and_name_line_segment_ray_line\"><\/span><strong data-start=\"1970\" data-end=\"1978\">Q11.<\/strong> Draw and name: line segment, ray, line.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"2021\" data-end=\"2070\"><strong data-start=\"2021\" data-end=\"2030\">Hint:<\/strong> Use ruler and pencil.<br data-start=\"2052\" data-end=\"2055\" \/><strong data-start=\"2055\" data-end=\"2068\">Solution:<\/strong><\/p>\n<ul data-start=\"2071\" data-end=\"2225\">\n<li data-start=\"2071\" data-end=\"2109\">\n<p data-start=\"2073\" data-end=\"2109\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mrow><mi><span class=\"mord mathnormal\">A<span style=\"font-size: revert; font-family: Manrope, sans-serif; letter-spacing: -0.1px;\">B<\/span><\/span><\/mi><\/mrow><mo stretchy=\"true\">\u203e<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\overline{AB}<\/annotation><\/semantics><\/math><\/span><\/span> \u2192 line segment<\/p>\n<\/li>\n<li data-start=\"2110\" data-end=\"2145\">\n<p data-start=\"2112\" data-end=\"2145\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mrow><mi>\n<\/mi><mi>\n<\/mi><\/mrow><\/mover><\/mrow><\/semantics><\/math><\/span><\/span><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><mo stretchy=\"true\">\u2192<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\overrightarrow{AB}<\/annotation><\/semantics><\/math><\/span><\/span> \u2192 ray<\/p>\n<\/li>\n<li data-start=\"2146\" data-end=\"2225\">\n<p data-start=\"2148\" data-end=\"2225\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><mo stretchy=\"true\">\u2194<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\overleftrightarrow{AB}<\/annotation><\/semantics><\/math><\/span><\/span> \u2192 line<br data-start=\"2184\" data-end=\"2187\" \/><strong data-start=\"2187\" data-end=\"2198\">Answer:<\/strong> As shown by symbols above.<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"2227\" data-end=\"2230\" \/>\n<h3 data-start=\"2232\" data-end=\"2293\"><span class=\"ez-toc-section\" id=\"Q12_Triangle_with_angles_90%C2%B0_and_45%C2%B0_%E2%86%92_find_third\"><\/span><strong data-start=\"2236\" data-end=\"2244\">Q12.<\/strong> Triangle with angles 90\u00b0 and 45\u00b0 \u2192 find third.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"2294\" data-end=\"2309\"><strong data-start=\"2294\" data-end=\"2307\">Solution:<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mtext>Sum<\/mtext><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><mo>\u21d2<\/mo><mn>90<\/mn><mo>+<\/mo><mn>45<\/mn><mo>+<\/mo><mi>x<\/mi><mo>=<\/mo><mn>180<\/mn><mo>\u21d2<\/mo><mi>x<\/mi><mo>=<\/mo><mn>45<\/mn><\/mrow><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"2390\" data-end=\"2429\"><strong data-start=\"2390\" data-end=\"2401\">Answer:<\/strong> Third angle = <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>45<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">45^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>.<\/p>\n<hr data-start=\"2431\" data-end=\"2434\" \/>\n<h3 data-start=\"2436\" data-end=\"2499\"><span class=\"ez-toc-section\" id=\"Q13_Two_angles_sum_130%C2%B0_%E2%86%92_find_supplementary_angles\"><\/span><strong data-start=\"2440\" data-end=\"2448\">Q13.<\/strong> Two angles sum 130\u00b0 \u2192 find supplementary angles.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"2500\" data-end=\"2553\"><strong data-start=\"2500\" data-end=\"2509\">Hint:<\/strong> Supplementary \u21d2 sum 180\u00b0.<br data-start=\"2535\" data-end=\"2538\" \/><strong data-start=\"2538\" data-end=\"2551\">Solution:<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>180<\/mn><mo>\u2212<\/mo><mn>130<\/mn><mo>=<\/mo><mn>50<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">180 &#8211; 130 = 50<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"2577\" data-end=\"2602\"><strong data-start=\"2577\" data-end=\"2588\">Answer:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>50<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">50^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>.<\/p>\n<hr data-start=\"2604\" data-end=\"2607\" \/>\n<h3 data-start=\"2609\" data-end=\"2668\"><span class=\"ez-toc-section\" id=\"Q14_Quadrilateral_angles_80%C2%B0_90%C2%B0_75%C2%B0_%E2%86%92_fourth\"><\/span><strong data-start=\"2613\" data-end=\"2621\">Q14.<\/strong> Quadrilateral angles 80\u00b0, 90\u00b0, 75\u00b0 \u2192 fourth?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>80<\/mn><mo>+<\/mo><mn>90<\/mn><mo>+<\/mo><mn>75<\/mn><mo>+<\/mo><mi>x<\/mi><mo>=<\/mo><mn>360<\/mn><\/mrow><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>\u21d2<\/mo><mi>x<\/mi><mo>=<\/mo><mn>115<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">80 + 90 + 75 + x = 360 \\Rightarrow x = 115<\/annotation><\/semantics><\/math><\/span><\/span><\/span><strong data-start=\"2720\" data-end=\"2731\">Answer:<\/strong> Fourth angle = <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>115<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">115^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>.<\/p>\n<hr data-start=\"2763\" data-end=\"2766\" \/>\n<h3 data-start=\"2768\" data-end=\"2818\"><span class=\"ez-toc-section\" id=\"Q15_Circle_radius_7_cm_%E2%86%92_circumference\"><\/span><strong data-start=\"2772\" data-end=\"2780\">Q15.<\/strong> Circle radius 7 cm \u2192 circumference.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mo>=<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mi>r<\/mi><mo>=<\/mo><mn>2<\/mn><mo>\u00d7<\/mo><mfrac><mn>22<\/mn><mn>7<\/mn><\/mfrac><mo>\u00d7<\/mo><mn>7<\/mn><mo>=<\/mo><mn>44<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">C = 2\\pi r = 2 \\times \\frac{22}{7} \\times 7 = 44<\/annotation><\/semantics><\/math><\/span><\/span><\/span><strong data-start=\"2876\" data-end=\"2887\">Answer:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>C<\/mi><mo>=<\/mo><mn>44<\/mn><mtext>\u00a0cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">C = 44\\ \\text{cm}<\/annotation><\/semantics><\/math><\/span><\/span>.<\/p>\n<hr data-start=\"2912\" data-end=\"2915\" \/>\n<h2 data-start=\"2917\" data-end=\"2962\"><span class=\"ez-toc-section\" id=\"Section_C_%E2%80%93_Application_3_Marks_Each\"><\/span><strong data-start=\"2920\" data-end=\"2962\">Section C \u2013 Application (3 Marks Each)<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 data-start=\"2964\" data-end=\"3028\"><span class=\"ez-toc-section\" id=\"Q16_Prove_Adjacent_angles_on_a_straight_line_180%C2%B0\"><\/span><strong data-start=\"2968\" data-end=\"2976\">Q16.<\/strong> Prove: Adjacent angles on a straight line = 180\u00b0.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"3029\" data-end=\"3156\"><strong data-start=\"3029\" data-end=\"3038\">Hint:<\/strong> Angles on straight line form linear pair.<br data-start=\"3080\" data-end=\"3083\" \/><strong data-start=\"3083\" data-end=\"3096\">Solution:<\/strong><br data-start=\"3096\" data-end=\"3099\" \/>Let \u22201 + \u22202 form straight line.<br data-start=\"3130\" data-end=\"3133\" \/>By linear-pair axiom:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u2220<\/mi><mn>1<\/mn><mo>+<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mn>2<\/mn><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\angle1 + \\angle2 = 180^\\circ<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"3195\" data-end=\"3220\"><strong data-start=\"3195\" data-end=\"3206\">Answer:<\/strong> Hence proved.<\/p>\n<hr data-start=\"3222\" data-end=\"3225\" \/>\n<h3 data-start=\"3227\" data-end=\"3292\"><span class=\"ez-toc-section\" id=\"Q17_Triangle_sides_6_cm_8_cm_10_cm_%E2%86%92_right_triangle\"><\/span><strong data-start=\"3231\" data-end=\"3239\">Q17.<\/strong> Triangle sides 6 cm, 8 cm, 10 cm \u2192 right triangle?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>6<\/mn><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mn>8<\/mn><mn>2<\/mn><\/msup><mo>=<\/mo><mn>36<\/mn><mo>+<\/mo><mn>64<\/mn><mo>=<\/mo><mn>100<\/mn><mo>=<\/mo><msup><mn>10<\/mn><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">6^2 + 8^2 = 36 + 64 = 100 = 10^2<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"3334\" data-end=\"3383\"><strong data-start=\"3334\" data-end=\"3345\">Answer:<\/strong> Yes, right-angled at side 6 and 8 cm.<\/p>\n<hr data-start=\"3385\" data-end=\"3388\" \/>\n<h3 data-start=\"3390\" data-end=\"3442\"><span class=\"ez-toc-section\" id=\"Q18_Circle_diameter_14_cm_%E2%86%92_radius_C_A\"><\/span><strong data-start=\"3394\" data-end=\"3402\">Q18.<\/strong> Circle diameter 14 cm \u2192 radius, C, A.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><mo>=<\/mo><mn>7<\/mn><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>C<\/mi><mo>=<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mi>r<\/mi><mo>=<\/mo><mn>44<\/mn><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>A<\/mi><mo>=<\/mo><mi>\u03c0<\/mi><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mfrac><mn>22<\/mn><mn>7<\/mn><\/mfrac><mo>\u00d7<\/mo><mn>7<\/mn><mo>\u00d7<\/mo><mn>7<\/mn><mo>=<\/mo><mn>154<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">r = 7,\\quad C = 2\\pi r = 44,\\quad A = \\pi r^2 = \\frac{22}{7}\\times7\\times7 = 154<\/annotation><\/semantics><\/math><\/span><\/span><\/span><strong data-start=\"3532\" data-end=\"3543\">Answer:<\/strong> Radius 7 cm, Circumference 44 cm, Area 154 cm\u00b2.<\/p>\n<hr data-start=\"3593\" data-end=\"3596\" \/>\n<h3 data-start=\"3598\" data-end=\"3664\"><span class=\"ez-toc-section\" id=\"Q19_Exterior_angle_120%C2%B0_interior_opposite_40%C2%B0_%E2%86%92_other\"><\/span><strong data-start=\"3602\" data-end=\"3610\">Q19.<\/strong> Exterior angle 120\u00b0, interior opposite 40\u00b0 \u2192 other?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>120<\/mn><mo>=<\/mo><mn>40<\/mn><mo>+<\/mo><mi>x<\/mi><mo>\u21d2<\/mo><mi>x<\/mi><mo>=<\/mo><mn>80<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">120 = 40 + x \\Rightarrow x = 80<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"3705\" data-end=\"3730\"><strong data-start=\"3705\" data-end=\"3716\">Answer:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>80<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">80^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>.<\/p>\n<h3 data-start=\"3737\" data-end=\"3793\"><span class=\"ez-toc-section\" id=\"Q20_In_parallelogram_ABCD_%E2%88%A0A_70%C2%B0_%E2%86%92_others\"><\/span><strong data-start=\"3741\" data-end=\"3749\">Q20.<\/strong> In parallelogram ABCD, \u2220A = 70\u00b0 \u2192 others?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u2220<\/mi><mi>A<\/mi><mo>=<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>C<\/mi><mo>=<\/mo><mn>70<\/mn><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi mathvariant=\"normal\">\u2220<\/mi><mi>B<\/mi><mo>=<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>D<\/mi><mo>=<\/mo><mn>110<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\angle A = \\angle C = 70,\\quad \\angle B = \\angle D = 110<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"3859\" data-end=\"3892\"><strong data-start=\"3859\" data-end=\"3870\">Answer:<\/strong> 70\u00b0, 110\u00b0, 70\u00b0, 110\u00b0.<\/p>\n<hr data-start=\"3894\" data-end=\"3897\" \/>\n<h2 data-start=\"3899\" data-end=\"3939\"><span class=\"ez-toc-section\" id=\"Section_D_%E2%80%93_HOTS_4%E2%80%935_Marks_Each\"><\/span><strong data-start=\"3902\" data-end=\"3939\">Section D \u2013 HOTS (4\u20135 Marks Each)<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 data-start=\"3941\" data-end=\"3984\"><span class=\"ez-toc-section\" id=\"Q21_Two_equal_angles_third_96%C2%B0\"><\/span><strong data-start=\"3945\" data-end=\"3953\">Q21.<\/strong> Two equal angles, third 96\u00b0.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mi>x<\/mi><mo>+<\/mo><mn>96<\/mn><mo>=<\/mo><mn>180<\/mn><mo>\u21d2<\/mo><mn>2<\/mn><mi>x<\/mi><mo>=<\/mo><mn>84<\/mn><mo>\u21d2<\/mo><mi>x<\/mi><mo>=<\/mo><mn>42<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + x + 96 = 180 \\Rightarrow 2x = 84 \\Rightarrow x = 42<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"4049\" data-end=\"4105\"><strong data-start=\"4049\" data-end=\"4060\">Answer:<\/strong> Equal angles = 42\u00b0 each; Isosceles triangle.<\/p>\n<hr data-start=\"4107\" data-end=\"4110\" \/>\n<h3 data-start=\"4112\" data-end=\"4158\"><span class=\"ez-toc-section\" id=\"Q22_Sum_of_interior_angles_1440%C2%B0\"><\/span><strong data-start=\"4116\" data-end=\"4124\">Q22.<\/strong> Sum of interior angles = 1440\u00b0.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"4159\" data-end=\"4193\">Formula \u2192 <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mn>180<\/mn><mo>=<\/mo><mn>1440<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(n-2) \u00d7 180 = 1440<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>8<\/mn><mo>\u21d2<\/mo><mi>n<\/mi><mo>=<\/mo><mn>10<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">n-2 = 8 \\Rightarrow n = 10<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"4229\" data-end=\"4268\"><strong data-start=\"4229\" data-end=\"4240\">Answer:<\/strong> 10-sided polygon (decagon).<\/p>\n<hr data-start=\"4270\" data-end=\"4273\" \/>\n<h3 data-start=\"4275\" data-end=\"4323\"><span class=\"ez-toc-section\" id=\"Q23_Circle_radius_4_cm_%E2%86%92_mark_points\"><\/span><strong data-start=\"4279\" data-end=\"4287\">Q23.<\/strong> Circle radius 4 cm \u2192 mark points.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"4324\" data-end=\"4422\"><strong data-start=\"4324\" data-end=\"4333\">Hint:<\/strong> Use compass radius 4 cm.<br data-start=\"4358\" data-end=\"4361\" \/><strong data-start=\"4361\" data-end=\"4372\">Answer:<\/strong> One inside, one on, one outside \u2014 as constructed.<\/p>\n<hr data-start=\"4424\" data-end=\"4427\" \/>\n<h3 data-start=\"4429\" data-end=\"4489\"><span class=\"ez-toc-section\" id=\"Q24_Quadrilateral_angles_110%C2%B0_95%C2%B0_70%C2%B0_%E2%86%92_fourth\"><\/span><strong data-start=\"4433\" data-end=\"4441\">Q24.<\/strong> Quadrilateral angles 110\u00b0, 95\u00b0, 70\u00b0 \u2192 fourth?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>110<\/mn><mo>+<\/mo><mn>95<\/mn><mo>+<\/mo><mn>70<\/mn><mo>+<\/mo><mi>x<\/mi><mo>=<\/mo><mn>360<\/mn><mo>\u21d2<\/mo><mi>x<\/mi><mo>=<\/mo><mn>85<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">110 + 95 + 70 + x = 360 \\Rightarrow x = 85<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"4541\" data-end=\"4597\"><strong data-start=\"4541\" data-end=\"4552\">Answer:<\/strong> Fourth angle = 85\u00b0; Irregular quadrilateral.<\/p>\n<hr data-start=\"4599\" data-end=\"4602\" \/>\n<h3 data-start=\"4604\" data-end=\"4647\"><span class=\"ez-toc-section\" id=\"Q25_AB8AB_8_cm_bisected_at_MAMMB824AM_MB_frac82_4\"><\/span><strong data-start=\"4608\" data-end=\"4616\">Q25.<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mo>=<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">AB = 8<\/annotation><\/semantics><\/math><\/span><\/span> cm bisected at M.<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><mi>M<\/mi><mo>=<\/mo><mi>M<\/mi><mi>B<\/mi><mo>=<\/mo><mfrac><mn>8<\/mn><mn>2<\/mn><\/mfrac><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">AM = MB = \\frac{8}{2} = 4<\/annotation><\/semantics><\/math><\/span><\/span><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"4682\" data-end=\"4724\"><strong data-start=\"4682\" data-end=\"4693\">Answer:<\/strong> AM = MB = 4 cm; M is midpoint.<\/p>\n<hr data-start=\"4726\" data-end=\"4729\" \/>\n<h2 data-start=\"4731\" data-end=\"4754\"><span class=\"ez-toc-section\" id=\"Section_E_%E2%80%93_MCQs\"><\/span><strong data-start=\"4734\" data-end=\"4754\">Section E \u2013 MCQs<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div class=\"_tableContainer_1rjym_1\">\n<div class=\"group _tableWrapper_1rjym_13 flex w-fit flex-col-reverse\" tabindex=\"-1\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"4756\" data-end=\"5055\">\n<thead data-start=\"4756\" data-end=\"4783\">\n<tr data-start=\"4756\" data-end=\"4783\">\n<th data-start=\"4756\" data-end=\"4762\" data-col-size=\"sm\">No.<\/th>\n<th data-start=\"4762\" data-end=\"4773\" data-col-size=\"sm\">Question<\/th>\n<th data-start=\"4773\" data-end=\"4783\" data-col-size=\"sm\">Answer<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"4799\" data-end=\"5055\">\n<tr data-start=\"4799\" data-end=\"4841\">\n<td data-start=\"4799\" data-end=\"4804\" data-col-size=\"sm\">26<\/td>\n<td data-col-size=\"sm\" data-start=\"4804\" data-end=\"4832\">Right angles in rectangle<\/td>\n<td data-col-size=\"sm\" data-start=\"4832\" data-end=\"4841\">D \u2013 4<\/td>\n<\/tr>\n<tr data-start=\"4842\" data-end=\"4902\">\n<td data-start=\"4842\" data-end=\"4847\" data-col-size=\"sm\">27<\/td>\n<td data-col-size=\"sm\" data-start=\"4847\" data-end=\"4883\">Angle between perpendicular lines<\/td>\n<td data-col-size=\"sm\" data-start=\"4883\" data-end=\"4902\">B \u2013 Right angle<\/td>\n<\/tr>\n<tr data-start=\"4903\" data-end=\"4945\">\n<td data-start=\"4903\" data-end=\"4908\" data-col-size=\"sm\">28<\/td>\n<td data-start=\"4908\" data-end=\"4933\" data-col-size=\"sm\">Sum of triangle angles<\/td>\n<td data-col-size=\"sm\" data-start=\"4933\" data-end=\"4945\">B \u2013 180\u00b0<\/td>\n<\/tr>\n<tr data-start=\"4946\" data-end=\"4996\">\n<td data-start=\"4946\" data-end=\"4951\" data-col-size=\"sm\">29<\/td>\n<td data-col-size=\"sm\" data-start=\"4951\" data-end=\"4982\">Centre to circumference line<\/td>\n<td data-col-size=\"sm\" data-start=\"4982\" data-end=\"4996\">C \u2013 Radius<\/td>\n<\/tr>\n<tr data-start=\"4997\" data-end=\"5055\">\n<td data-start=\"4997\" data-end=\"5002\" data-col-size=\"sm\">30<\/td>\n<td data-col-size=\"sm\" data-start=\"5002\" data-end=\"5020\">False statement<\/td>\n<td data-col-size=\"sm\" data-start=\"5020\" data-end=\"5055\">B \u2013 Every rectangle is a square<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<hr data-start=\"5057\" data-end=\"5060\" \/>\n<h2 data-start=\"5062\" data-end=\"5100\"><span class=\"ez-toc-section\" id=\"Bonus_Challenge_%E2%80%93_Clock_Problem\"><\/span><strong data-start=\"5065\" data-end=\"5100\">Bonus Challenge \u2013 Clock Problem<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"63\" data-end=\"219\">At 3:00 the clock hands are at a right angle (90\u00b0). Find the next time between 3:00 and 4:00 when they again form a 90\u00b0 angle.<\/p>\n<hr data-start=\"221\" data-end=\"224\" \/>\n<h3 data-start=\"226\" data-end=\"309\"><span class=\"ez-toc-section\" id=\"1_Write_angles_of_hour_and_minute_hands_in_degrees_at_time_3_t3_t_minutes\"><\/span>1) Write angles of hour and minute hands (in degrees) at time <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><mo lspace=\"0em\" rspace=\"0em\">:<\/mo><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">3{:}t<\/annotation><\/semantics><\/math><\/span><\/span> minutes<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul data-start=\"311\" data-end=\"417\">\n<li data-start=\"311\" data-end=\"417\">\n<p data-start=\"313\" data-end=\"417\">Hour hand: at 3:00 it is at <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>90<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">90^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>. It moves <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>0.5<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">0.5^\\circ<\/annotation><\/semantics><\/math><\/span><\/span> per minute, so at <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">t<\/annotation><\/semantics><\/math><\/span><\/span> minutes past 3:<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"5102\" data-end=\"5223\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mtext>Hour\u00a0angle<\/mtext><mo>=<\/mo><mn>90<\/mn><mo>+<\/mo><mn>0.5<\/mn><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Hour angle} = 90 + 0.5t<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<ul data-start=\"455\" data-end=\"520\">\n<li data-start=\"455\" data-end=\"520\">\n<p data-start=\"457\" data-end=\"520\">Minute hand: moves <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>6<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">6^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>per minute, so at <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">t<\/annotation><\/semantics><\/math><\/span><\/span> minutes:<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"5102\" data-end=\"5223\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mtext>Minute\u00a0angle<\/mtext><mo>=<\/mo><mn>6<\/mn><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Minute angle} = 6t<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<h3 data-start=\"553\" data-end=\"583\"><span class=\"ez-toc-section\" id=\"2_Angle_between_the_hands\"><\/span>2) Angle between the hands<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"585\" data-end=\"645\">The (smaller) angle between them is the absolute difference:<\/p>\n<p data-start=\"5102\" data-end=\"5223\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mtext>Angle<\/mtext><mo>=<\/mo><mo fence=\"false\" maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">\u2223<\/mo><mn>6<\/mn><mi>t<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>90<\/mn><mo>+<\/mo><mn>0.5<\/mn><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo fence=\"false\" maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">\u2223<\/mo><mo>=<\/mo><mrow><mo fence=\"true\">\u2223<\/mo><mfrac><mn>11<\/mn><mn>2<\/mn><\/mfrac><mi>t<\/mi><mo>\u2212<\/mo><mn>90<\/mn><mo fence=\"true\">\u2223<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Angle} = \\big|6t &#8211; (90 + 0.5t)\\big| = \\left|\\frac{11}{2}t &#8211; 90\\right|<\/annotation><\/semantics><\/math><\/span><\/span><\/span>(we used <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>6<\/mn><mi>t<\/mi><mo>\u2212<\/mo><mn>0.5<\/mn><mi>t<\/mi><mo>=<\/mo><mn>5.5<\/mn><mi>t<\/mi><mo>=<\/mo><mfrac><mn>11<\/mn><mn>2<\/mn><\/mfrac><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">6t &#8211; 0.5t = 5.5t = \\tfrac{11}{2}t<\/annotation><\/semantics><\/math><\/span><\/span>).<\/p>\n<h3 data-start=\"778\" data-end=\"830\"><span class=\"ez-toc-section\" id=\"3_Set_the_angle_equal_to_90%E2%88%9890circ90%E2%88%98_and_solve\"><\/span>3) Set the angle equal to <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>90<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">90^\\circ<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">9<\/span><span class=\"mord\">0<span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> and solve<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"832\" data-end=\"839\">We need<\/p>\n<p data-start=\"5102\" data-end=\"5223\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mrow><mo fence=\"true\">\u2223<\/mo><mfrac><mn>11<\/mn><mn>2<\/mn><\/mfrac><mi>t<\/mi><mo>\u2212<\/mo><mn>90<\/mn><mo fence=\"true\">\u2223<\/mo><\/mrow><mo>=<\/mo><mn>90.<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\left|\\frac{11}{2}t &#8211; 90\\right| = 90.<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"884\" data-end=\"905\">This gives two cases.<\/p>\n<p data-start=\"907\" data-end=\"947\"><strong data-start=\"907\" data-end=\"918\">Case A:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mn>11<\/mn><mn>2<\/mn><\/mfrac><\/mstyle><mi>t<\/mi><mo>\u2212<\/mo><mn>90<\/mn><mo>=<\/mo><mn>90<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\dfrac{11}{2}t &#8211; 90 = 90<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<ul data-start=\"949\" data-end=\"1800\">\n<li data-start=\"949\" data-end=\"1010\">\n<p data-start=\"951\" data-end=\"976\">Add <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>90<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">90<\/annotation><\/semantics><\/math><\/span><\/span>\u00a0to both sides:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mn>11<\/mn><mn>2<\/mn><\/mfrac><mi>t<\/mi><mo>=<\/mo><mn>180.<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\dfrac{11}{2}t = 180.<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/li>\n<li data-start=\"1011\" data-end=\"1065\">\n<p data-start=\"1013\" data-end=\"1042\">Multiply both sides by <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2<\/annotation><\/semantics><\/math><\/span><\/span>:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>11<\/mn><mi>t<\/mi><mo>=<\/mo><mn>360.<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">11t = 360.<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/li>\n<li data-start=\"1066\" data-end=\"1325\">\n<p data-start=\"1068\" data-end=\"1117\">Divide by <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>11<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">11<\/annotation><\/semantics><\/math><\/span><\/span>. Do the division digit-by-digit:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>t<\/mi><mo>=<\/mo><mfrac><mn>360<\/mn><mn>11<\/mn><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">t = \\frac{360}{11}.<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"1152\" data-end=\"1274\">Long division: <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>11<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">11<\/annotation><\/semantics><\/math><\/span><\/span> goes into <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>360<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">360<\/annotation><\/semantics><\/math><\/span><\/span> exactly <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>32<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">32<\/annotation><\/semantics><\/math><\/span><\/span> times (because <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>11<\/mn><mo>\u00d7<\/mo><mn>32<\/mn><mo>=<\/mo><mn>352<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">11\\times32=352<\/annotation><\/semantics><\/math><\/span><\/span>), remainder <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>360<\/mn><mo>\u2212<\/mo><mn>352<\/mn><mo>=<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">360-352=8<\/annotation><\/semantics><\/math><\/span><\/span>.<br data-start=\"1267\" data-end=\"1270\" \/>So<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>t<\/mi><mo>=<\/mo><mn>32<\/mn><mtext>\u2005\u200a<\/mtext><mfrac><mn>8<\/mn><mn>11<\/mn><\/mfrac><mtext>\u00a0minutes<\/mtext><mi mathvariant=\"normal\">.<\/mi><\/mrow><\/semantics><\/math><\/span><\/span><\/span><\/li>\n<li data-start=\"1327\" data-end=\"1800\">\n<p data-start=\"1329\" data-end=\"1388\">Convert the fractional minute <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mn>8<\/mn><mn>11<\/mn><\/mfrac><\/mstyle><\/mrow><annotation encoding=\"application\/x-tex\">\\dfrac{8}{11}<\/annotation><\/semantics><\/math><\/span><\/span> to seconds:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mn>8<\/mn><mn>11<\/mn><\/mfrac><mo>\u00d7<\/mo><mn>60<\/mn><mo>=<\/mo><mfrac><mn>480<\/mn><mn>11<\/mn><\/mfrac><mtext>\u00a0seconds<\/mtext><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{8}{11}\\times 60 = \\frac{480}{11}\\ \\text{seconds}.<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"1459\" data-end=\"1582\">Divide <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>480<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">480<\/annotation><\/semantics><\/math><\/span><\/span> by <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>11<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">11<\/annotation><\/semantics><\/math><\/span><\/span> : <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>11<\/mn><mo>\u00d7<\/mo><mn>43<\/mn><mo>=<\/mo><mn>473<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">11\\times43=473<\/annotation><\/semantics><\/math><\/span><\/span>, remainder <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>480<\/mn><mo>\u2212<\/mo><mn>473<\/mn><mo>=<\/mo><mn>7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">480-473=7<\/annotation><\/semantics><\/math><\/span><\/span>.<br data-start=\"1529\" data-end=\"1532\" \/>So <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mn>480<\/mn><mn>11<\/mn><\/mfrac><\/mstyle><mo>=<\/mo><mn>43<\/mn><mtext>\u2005\u200a<\/mtext><mfrac><mn>7<\/mn><mn>11<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\dfrac{480}{11}=43\\;\\frac{7}{11}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> seconds.<\/p>\n<p data-start=\"1586\" data-end=\"1595\">Therefore<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>t<\/mi><mo>=<\/mo><mn>32<\/mn><mtext>\u00a0minutes<\/mtext><mtext>\u2005\u200a<\/mtext><mn>43<\/mn><mtext>\u2005\u200a<\/mtext><mfrac><mn>7<\/mn><mn>11<\/mn><\/mfrac><mtext>\u00a0seconds<\/mtext><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">t = 32\\ \\text{minutes}\\;43\\;\\frac{7}{11}\\ \\text{seconds}.<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">.<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"1669\" data-end=\"1734\">As a decimal, <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mn>7<\/mn><mn>11<\/mn><\/mfrac><\/mstyle><\/mrow><annotation encoding=\"application\/x-tex\">\\dfrac{7}{11}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">117<\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> second <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>\u2248<\/mo><mn>0.63636<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\approx 0.63636<\/annotation><\/semantics><\/math><\/span><\/span> s, so<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>t<\/mi><mo>\u2248<\/mo><mn>32<\/mn><mtext>\u00a0minutes\u00a0<\/mtext><mn>43.636<\/mn><mtext>\u00a0seconds<\/mtext><mi mathvariant=\"normal\">.<\/mi><\/mrow><\/semantics><\/math><\/span><\/span><\/span><\/li>\n<\/ul>\n<p data-start=\"1802\" data-end=\"1843\"><strong data-start=\"1802\" data-end=\"1813\">Case B:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mn>11<\/mn><mn>2<\/mn><\/mfrac><\/mstyle><mi>t<\/mi><mo>\u2212<\/mo><mn>90<\/mn><mo>=<\/mo><mo>\u2212<\/mo><mn>90<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\dfrac{11}{2}t &#8211; 90 = -90<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<ul data-start=\"1845\" data-end=\"1922\">\n<li data-start=\"1845\" data-end=\"1922\">\n<p data-start=\"1847\" data-end=\"1872\">Add <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>90<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">90<\/annotation><\/semantics><\/math><\/span><\/span> to both sides:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mn>11<\/mn><mn>2<\/mn><\/mfrac><mi>t<\/mi><mo>=<\/mo><mn>0<\/mn><mo>\u21d2<\/mo><mi>t<\/mi><mo>=<\/mo><mn>0.<\/mn><\/mrow><\/semantics><\/math><\/span><\/span><\/span><\/li>\n<\/ul>\n<p data-start=\"1923\" data-end=\"1986\">This is the starting time <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><mo lspace=\"0em\" rspace=\"0em\">:<\/mo><mn>00<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3{:}00<\/annotation><\/semantics><\/math><\/span><\/span> (the initial right angle).<\/p>\n<p data-start=\"1988\" data-end=\"2069\">We want the <em data-start=\"2000\" data-end=\"2006\">next<\/em> time after 3:00, so we take the positive solution from Case A.<\/p>\n<hr data-start=\"2071\" data-end=\"2074\" \/>\n<h3 data-start=\"2076\" data-end=\"2116\"><span class=\"ez-toc-section\" id=\"Final_answer_exact_and_approximate\"><\/span>Final answer (exact and approximate)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul data-start=\"2118\" data-end=\"2300\">\n<li data-start=\"2118\" data-end=\"2227\">\n<p data-start=\"2120\" data-end=\"2227\">Exact: <strong data-start=\"2127\" data-end=\"2173\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>t<\/mi><mo>=<\/mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mn>360<\/mn><mn>11<\/mn><\/mfrac><\/mstyle><\/mrow><annotation encoding=\"application\/x-tex\">t = \\dfrac{360}{11}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> minutes after 3:00<\/strong>, i.e. <strong data-start=\"2180\" data-end=\"2224\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><mo lspace=\"0em\" rspace=\"0em\">:<\/mo><mn>32<\/mn><mtext>\u2009\u2063<\/mtext><mo>:<\/mo><mtext>\u2009\u2063<\/mtext><mn>43<\/mn><mtext>\u00a0<\/mtext><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mn>7<\/mn><mn>11<\/mn><\/mfrac><\/mstyle><\/mrow><annotation encoding=\"application\/x-tex\">3{:}32\\!:\\!43\\ \\dfrac{7}{11}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> seconds<\/strong>.<\/p>\n<\/li>\n<li data-start=\"2228\" data-end=\"2300\">\n<p data-start=\"2230\" data-end=\"2300\">Approximate: <strong data-start=\"2243\" data-end=\"2261\">3:32:43.636&#8230;<\/strong> (3 hours, 32 minutes, 43.636 seconds).<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"2302\" data-end=\"2363\">So the hands next form a right angle at <strong data-start=\"2342\" data-end=\"2362\">about 3:32:43.64<\/strong>.<\/p>\n\n    <div class=\"xs_social_share_widget xs_share_url after_content \t\tmain_content  wslu-style-1 wslu-share-box-shaped wslu-fill-colored wslu-none wslu-share-horizontal wslu-theme-font-no wslu-main_content\">\n\n\t\t\n        <ul>\n\t\t\t        <\/ul>\n    <\/div> \n","protected":false},"excerpt":{"rendered":"<div class='heateorSssClear'><\/div><div  class='heateor_sss_sharing_container heateor_sss_horizontal_sharing' data-heateor-sss-href='https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/basics-of-geometry-notes\/'><div class='heateor_sss_sharing_title' style=\"font-weight:bold\" >Share\/Assign<\/div><div class=\"heateor_sss_sharing_ul\"><a aria-label=\"Facebook\" class=\"heateor_sss_facebook\" 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Introduction to Geometry Geometry is the branch of mathematics that deals with shapes, sizes, relative positions of figures, and properties of space. The word Geometry comes from Greek words: Geo = Earth and Metron = Measurement. 2. Fundamental Terms 2.1 Point A point is a precise location in space. 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Introduction to Geometry Geometry is the branch of mathematics that deals with shapes, sizes, relative positions of figures, and properties of space. The word Geometry comes from Greek words: Geo = Earth and Metron = Measurement. 2. Fundamental Terms 2.1 Point A point is a precise location in space. 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