1. 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.

• It has no length, breadth, or thickness.

• Notation: Usually denoted by a capital letter, e.g., $A, B, C$.

2.2 Line

• A line is a straight path of points extending infinitely in both directions.

• Notation: Line passing through points $A$ and $B$ is written as $AB$ or $\overleftrightarrow{AB}$.

2.3 Line Segment

• A line segment is part of a line with two endpoints.

• Notation: Line segment joining points $A$ and $B$ is $\overline{AB}$.

2.4 Ray

• A ray starts at one point and extends infinitely in one direction.

• Notation: Ray starting at $A$ passing through $B$ is $\overrightarrow{AB}$.

2.5 Plane

• A plane is a flat surface that extends infinitely in all directions.

• Represented by a quadrilateral figure in diagrams.

3. Angle And Its Types

• An angle is formed by two rays with a common endpoint called the vertex.

3.1 Classification

1. Acute Angle: $0^\circ < \theta < 90^\circ$
   

2. Right Angle: $\theta = 90^\circ$
   

3. Obtuse Angle: $90^\circ < \theta < 180^\circ$
   

4. Straight Angle: $\theta = 180^\circ$
   

5. Reflex Angle: $180^\circ < \theta < 360^\circ$
   

Formula:

$$\text{Sum of angles on a straight line} = 180^\circ$$$$\text{Sum of angles around a point} = 360^\circ$$

4. Triangle And Its Types

• A triangle has 3 sides and 3 angles.

4.1 Based on Sides

1. Equilateral: All sides equal

2. Isosceles: Two sides equal

3. Scalene: All sides unequal

4.2 Based on Angles

1. Acute-angled: All angles < $90^\circ$
   

2. Right-angled: One angle = $90^\circ$
   

3. Obtuse-angled: One angle > $90^\circ$
   

Important Property (Triangle Sum Theorem):

$$\text{Sum of angles in a triangle} = 180^\circ$$

5. Quadrilateral And Its Types

• A quadrilateral has 4 sides and 4 angles.

5.1 Types

1. Square: All sides equal, all angles $90^\circ$
   

2. Rectangle: Opposite sides equal, all angles $90^\circ$
   

3. Parallelogram: Opposite sides parallel and equal

4. Rhombus: All sides equal, opposite angles equal

5. Trapezium: One pair of opposite sides parallel

Property: Sum of angles in a quadrilateral:

$$\text{Sum of angles} = 360^\circ$$

6. Circles And Its Parts.

• A circle is a set of points equidistant from a fixed point called the centre.

• Radius (r): Distance from centre to any point on the circle

• Diameter (d): Twice the radius, $d = 2r$
  

• Circumference (C): $C = 2 \pi r$
  

• Area (A): $A = \pi r^2$
  

7. Basic Geometrical Constructions

1. Constructing a bisector of a line segment

2. Constructing an angle bisector

3. Constructing perpendiculars from a point on a line or outside a line

4. Constructing triangles using SSS, SAS, ASA, RHS criteria

8. Important Theorems

1. Pythagoras Theorem:

$$\text{In a right triangle: } AB^2 + BC^2 = AC^2$$

2. Triangle Sum Theorem:

$$\angle A + \angle B + \angle C = 180^\circ$$

3. Exterior Angle Theorem:

$$\text{Exterior angle of a triangle} = \text{Sum of the two opposite interior angles}$$

9. Tips & Tricks

• Always label points clearly in diagrams.

• Use a protractor for accurate angle measurement.

• Remember the sum of angles for triangle = 180°, quadrilateral = 360°.

• Practice constructing triangles using different combinations of sides and angles.

Diagram Placeholders:

• [Point, Line, Line Segment, Ray]

• [Triangle with labeled angles]

• [Quadrilateral types]

• [Circle with radius and diameter]

Worksheet on Basics of Geometry (Math Olympiad Preparation)

Topic: Basics of Geometry
Class: 6–9 (CBSE & Olympiad Level)
Marks: Practice Worksheet
Time: 45–60 minutes

Section A — Very Short Answer Questions (1 Mark Each)

(Concept Recall & Definitions)

1. Define a point and a line in geometry.

2. How many endpoints does a line segment have?

3. What is the measure of a straight angle?

4. Name the instrument used to draw circles.

5. Write the sum of all angles around a point.

6. How many vertices does a triangle have?

7. What is the sum of the angles of a quadrilateral?

8. Which type of angle is greater than 180° but less than 360°?

9. Write the relationship between radius and diameter.

10. Give one example of a real-life object in the shape of a circle.

Section B — Short Answer Questions (2 Marks Each)

(Understanding & Application)

11. Draw and name the following:
     (a) Line segment
     (b) Ray
     (c) Line

12. If one angle of a triangle is $90^\circ$ and another is $45^\circ$, find the third angle.

13. The sum of two angles is $130^\circ$. Find the measure of their supplementary angles.

14. In a quadrilateral, three angles are $80^\circ, 90^\circ, 75^\circ$. Find the fourth angle.

15. A circle has a radius of 7 cm. Find its circumference using $\pi = \frac{22}{7}$.
     $C=2\pi r=2\times \frac{22}{7}\times 7=?$

Section C — Application / Reasoning (3 Marks Each)

16. The sum of two adjacent angles on a straight line is always $180^\circ$. Prove this statement using a neat diagram.

17. A triangle has sides 6 cm, 8 cm, and 10 cm. Verify whether it is a right-angled triangle using the Pythagoras Theorem.
     $a^2 + b^2 = c^2$
    

18. In a circle with diameter 14 cm, find:
     (a) Radius
     (b) Circumference
     (c) Area
     Use $\pi = \frac{22}{7}$.

19. The exterior angle of a triangle is $120^\circ$ and one of the interior opposite angles is $40^\circ$.
     Find the other interior opposite angle.

20. In parallelogram ABCD, $\angle A = 70^\circ$. Find all other angles.
     (Hint: Opposite angles are equal and adjacent angles are supplementary.)

Section D — Higher Order Thinking (HOTS) Questions (4–5 Marks Each)

21. A triangle has two equal angles and the third angle is $96^\circ$.
     Find each of the equal angles and name the triangle.

22. The sum of the interior angles of an $n$-sided polygon is $1440^\circ$.
     Find the value of $n$.

23. Draw a circle of radius 4 cm, mark points:
     (a) Inside the circle
     (b) On the circle
     (c) Outside the circle

24. A quadrilateral has three angles measuring $110^\circ, 95^\circ, 70^\circ$. Find the fourth angle and classify the quadrilateral.

25. A line segment $AB = 8 \text{ cm}$ is bisected at $M$.
     Find $AM$ and $MB$, and justify your answer using geometric reasoning.

Section E — Multiple Choice Questions (MCQs)

(For Quick Revision)

26. The total number of right angles in a rectangle is:
     A. 1 B. 2 C. 3 D. 4

27. The angle formed by two perpendicular lines is:
     A. Acute B. Right C. Obtuse D. Straight

28. The sum of all angles in a triangle is: 
     A. $90^\circ$ B. $180^\circ$ C. $270^\circ$ D. $360^\circ$
    

29. A line segment joining the centre of a circle to a point on its circumference is called:
     A. Chord B. Diameter C. Radius D. Tangent

30. Which of the following statements is false?
     A. Every square is a rectangle.
     B. Every rectangle is a square.
     C. Every square is a rhombus.
     D. Every rhombus is a parallelogram.

Worksheet Hints, Solutions & Answers

Section A – Very Short Answer (1 Mark Each)

Q1. Define a point and a line.

Hint: Recall definitions from basic geometry.
Solution:

• A point has position but no size.

• A line extends endlessly in both directions, made of infinite points.
  Answer: Point – no dimensions; Line – extends infinitely both ways.

Q2. How many endpoints does a line segment have?

Hint: Think of a line with fixed ends.
Solution: A line segment has two endpoints.
Answer: 2 endpoints.

Q3. Measure of a straight angle?

Hint: It lies on a straight line.
Solution: $180^\circ$.
Answer: $180^\circ$

Q4. Instrument to draw circles?

Hint: Used with pencil and pin.
Answer: Compass.

Q5. Sum of all angles around a point?

Solution:

$$\text{Sum of all angles around a point} = 360^\circ$$

Answer: $360^\circ$

Q6. Vertices of a triangle?

Answer: 3 vertices.

Q7. Sum of angles of a quadrilateral?

Solution:

$$\text{Sum} = 360^\circ$$

Answer: $360^\circ$

Q8. Type of angle greater than 180° but less than 360°?

Answer: Reflex angle.

Q9. Relationship between radius (r) and diameter (d)?

$$d = 2r$$Answer: Diameter = 2 × Radius.

Q10. Example of circle shape?

Answer: Clock, coin, wheel.

Section B – Short Answer (2 Marks Each)

Q11. Draw and name: line segment, ray, line.

Hint: Use ruler and pencil.
Solution:

• $\overline{AB}$ → line segment

• $\stackrel{}{}$$\overrightarrow{AB}$ → ray

• $\overleftrightarrow{AB}$ → line
  Answer: As shown by symbols above.

Q12. Triangle with angles 90° and 45° → find third.

Solution:

$$\text{Sum}={180}^{\circ }\Rightarrow 90+45+x=180\Rightarrow x=45$$

Answer: Third angle = $45^\circ$.

Q13. Two angles sum 130° → find supplementary angles.

Hint: Supplementary ⇒ sum 180°.
Solution:

$$180 – 130 = 50$$

Answer: $50^\circ$.

Q14. Quadrilateral angles 80°, 90°, 75° → fourth?

$$80+90+75+x=360$$

$$80 + 90 + 75 + x = 360 \Rightarrow x = 115$$Answer: Fourth angle = $115^\circ$.

Q15. Circle radius 7 cm → circumference.

$$C = 2\pi r = 2 \times \frac{22}{7} \times 7 = 44$$Answer: $C = 44\ \text{cm}$.

Section C – Application (3 Marks Each)

Q16. Prove: Adjacent angles on a straight line = 180°.

Hint: Angles on straight line form linear pair.
Solution:
Let ∠1 + ∠2 form straight line.
By linear-pair axiom:

$$\angle1 + \angle2 = 180^\circ$$

Answer: Hence proved.

Q17. Triangle sides 6 cm, 8 cm, 10 cm → right triangle?

$$6^2 + 8^2 = 36 + 64 = 100 = 10^2$$

Answer: Yes, right-angled at side 6 and 8 cm.

Q18. Circle diameter 14 cm → radius, C, A.

$$r = 7,\quad C = 2\pi r = 44,\quad A = \pi r^2 = \frac{22}{7}\times7\times7 = 154$$Answer: Radius 7 cm, Circumference 44 cm, Area 154 cm².

Q19. Exterior angle 120°, interior opposite 40° → other?

$$120 = 40 + x \Rightarrow x = 80$$

Answer: $80^\circ$.

Q20. In parallelogram ABCD, ∠A = 70° → others?

$$\angle A = \angle C = 70,\quad \angle B = \angle D = 110$$

Answer: 70°, 110°, 70°, 110°.

Section D – HOTS (4–5 Marks Each)

Q21. Two equal angles, third 96°.

$$x + x + 96 = 180 \Rightarrow 2x = 84 \Rightarrow x = 42$$

Answer: Equal angles = 42° each; Isosceles triangle.

Q22. Sum of interior angles = 1440°.

Formula → $(n-2) × 180 = 1440$

$$n-2 = 8 \Rightarrow n = 10$$

Answer: 10-sided polygon (decagon).

Q23. Circle radius 4 cm → mark points.

Hint: Use compass radius 4 cm.
Answer: One inside, one on, one outside — as constructed.

Q24. Quadrilateral angles 110°, 95°, 70° → fourth?

$$110 + 95 + 70 + x = 360 \Rightarrow x = 85$$

Answer: Fourth angle = 85°; Irregular quadrilateral.

Q25. $AB = 8$ cm bisected at M.$$AM = MB = \frac{8}{2} = 4$$

Answer: AM = MB = 4 cm; M is midpoint.

Section E – MCQs

Bonus Challenge – Clock Problem

At 3:00 the clock hands are at a right angle (90°). Find the next time between 3:00 and 4:00 when they again form a 90° angle.

1) Write angles of hour and minute hands (in degrees) at time $3{:}t$ minutes

• Hour hand: at 3:00 it is at $90^\circ$. It moves $0.5^\circ$ per minute, so at $t$ minutes past 3:

$$\text{Hour angle} = 90 + 0.5t$$

• Minute hand: moves $6^\circ$per minute, so at $t$ minutes:

$$\text{Minute angle} = 6t$$

2) Angle between the hands

The (smaller) angle between them is the absolute difference:

$$\text{Angle} = \big|6t – (90 + 0.5t)\big| = \left|\frac{11}{2}t – 90\right|$$(we used $6t – 0.5t = 5.5t = \tfrac{11}{2}t$).

3) Set the angle equal to $90^\circ$ and solve

We need

$$\left|\frac{11}{2}t – 90\right| = 90.$$

This gives two cases.

Case A: $\dfrac{11}{2}t – 90 = 90$

• Add $90$ to both sides:

  $$\dfrac{11}{2}t = 180.$$

• Multiply both sides by $2$:

  $$11t = 360.$$

• Divide by $11$. Do the division digit-by-digit:

  $$t = \frac{360}{11}.$$

  Long division: $11$ goes into $360$ exactly $32$ times (because $11\times32=352$), remainder $360-352=8$.
  So

  $$t=32\frac{8}{11}\text{ minutes}\mathrm{.}$$

• Convert the fractional minute $\dfrac{8}{11}$ to seconds:

  $$\frac{8}{11}\times 60 = \frac{480}{11}\ \text{seconds}.$$
  

  Divide $480$ by $11$ : $11\times43=473$, remainder $480-473=7$.
  So $\dfrac{480}{11}=43\;\frac{7}{11}$ seconds.

  Therefore

  $$t = 32\ \text{minutes}\;43\;\frac{7}{11}\ \text{seconds}.$$

  As a decimal, $\dfrac{7}{11}$ second $\approx 0.63636$ s, so

  $$t\approx 32\text{ minutes }43.636\text{ seconds}\mathrm{.}$$

Case B: $\dfrac{11}{2}t – 90 = -90$

• Add $90$ to both sides:

  $$\frac{11}{2}t=0\Rightarrow t=0.$$

This is the starting time $3{:}00$ (the initial right angle).

We want the next time after 3:00, so we take the positive solution from Case A.

Final answer (exact and approximate)

• Exact: $t = \dfrac{360}{11}$ minutes after 3:00, i.e. $3{:}32\!:\!43\ \dfrac{7}{11}$ seconds.

• Approximate: 3:32:43.636… (3 hours, 32 minutes, 43.636 seconds).

So the hands next form a right angle at about 3:32:43.64.