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CBSE 9 | Mathematics
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LINES AND ANGLES | Study
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LINES AND ANGLES | Study
Mind Map Overal Idea Content Speed Notes Quick Coverage Basic terms and Definitions 1. Point – A Point is that which has no component. It is represented by a dot. 2. Line – When we join two distinct points then we get a line. A line has no endpoints; it can be extended on both readmore
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Content
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Basic terms and Definitions
1. Point – A Point is that which has no component. It is represented by a dot.
2. Line – When we join two distinct points then we get a line. A line has no endpoints; it can be extended on both sides infinitely.
3. Line Segment Line – Segment is the part of the line which has two endpoints.
4. Ray – Ray is also a part of the line that has only one endpoint and has no end on the other side.
5. Collinear points: Points lying on the same line are called Collinear Points.
6. Non-collinear points: Points which do not lie on the same line are called Non-Collinear Points. (Scroll to continue …)
(Scroll down till end of the page)
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Angles
When two rays begin from the same endpoint then they form an Angle. The two rays are the arms of the angle and the endpoint is the vertex of the angle.
Types of Angles
Angle Notation Image Acute An angle which is between 0° and 90°. Right An angle which is exactly equal to 90°. Obtuse An angle which is between 90° and 180°. Reflex An angle which is between 180° and 360° Straight An angle which is exactly equal to 180°. Complete An angle which is exactly equal to 360°. Types of Angles Complementary and Supplementary Angles
Complementary Angles are the different angles whose sum is 90°.
Complementary Angles are the different angles whose sum is 180°.
Intersecting Lines and Non-intersecting Lines
There are two ways to draw two lines-
1. The lines which cross each other from a particular point are called Intersecting Lines.
2. The lines which never cross each other at any point are called Non-intersecting Lines. These lines are called Parallel Lines and the common length between two lines is the distance between parallel lines.
Pairs of Angles Axioms
1. If a ray stands on a line, then the sum of two adjacent angles formed by that ray is 180°.
This shows that the common arm of the two angles is the ray which is standing on a line and the two adjacent angles are the linear pair of the angles. As the sum of two angles is 180° so these are supplementary angles too.
2. If the sum of two adjacent angles is 180°, then the arms which are not common of the angles form a line.
This is the reverse of the first axiom which says that the opposite is also true.
Vertically opposite Angles Theorem
When two lines intersect each other, then the vertically opposite angles so formed will be equal.
AC and BD are intersecting each other so ∠AOD = ∠BOC and ∠AOB = DOC.
Parallel Lines and a Transversal
If a line passes through two distinct lines and intersects them at distant points then this line is called Transversal Line.
Here line “l” is transversal of line m and n.
Exterior Angles – ∠1, ∠2, ∠7 and ∠8
Interior Angles – ∠3, ∠4, ∠5 and ∠6
Pairs of angles formed when a transversal intersects two lines-
1. Corresponding Angles:
- ∠ 1 and ∠ 5
- ∠ 2 and ∠ 6
- ∠ 4 and ∠ 8
- ∠ 3 and ∠ 7
2. Alternate Interior Angles:
- ∠ 4 and ∠ 6
- ∠ 3 and ∠ 5
3. Alternate Exterior Angles:
- ∠ 1 and ∠ 7
- ∠ 2 and ∠ 8
4. Interior Angles on the same side of the transversal:
- ∠ 4 and ∠ 5
- ∠ 3 and ∠ 6
Transversal Axioms
1. If a transversal intersects two parallel lines, then
- Each pair of corresponding angles will be equal.
- Each pair of alternate interior angles will be equal.
- Each pair of interior angles on the same side of the transversal will be supplementary.
2. If a transversal intersects two lines in such a way that
- Corresponding angles are equal then these two lines will be parallel to each other.
- Alternate interior angles are equal then the two lines will be parallel.
- Interior angles on the same side of the transversal are supplementary then the two lines will be parallel.
Lines Parallel to the Same Line
If two lines are parallel with a common line then these two lines will also be parallel to each other.
As in the above figure if AB ∥ CD and EF ∥ CD then AB ∥ EF.
Angle Sum Property of a Triangle
1. The sum of the angles of a triangle is 180º.
∠A + ∠B + ∠C = 180°
2. If we produce any side of a triangle, then the exterior angle formed is equal to the sum of the two interior opposite angles.
∠BCD = ∠BAC + ∠ABC.
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Advanced Tools For Study Assess Interact TRIANGLES | Study
Mind Map Overal Idea Content Speed Notes Quick Coverage Similar Geometric Figures: Two geometric figures which are the same in shape, such that one is simply a copy of the other on a smaller scale or a larger scale, are called similar geometric figures. Two geometric figures are said to be similar if and only readmore
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Similar Geometric Figures:
Two geometric figures which are the same in shape, such that one is simply a copy of the other on a smaller scale or a larger scale, are called similar geometric figures.
Two geometric figures are said to be similar if and only if they have the same shape but not necessarily the same size. Two congruent geometric figures are always similar but converse may or may not be true. (Scroll down till end of the page)
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Similar Polygons: Two polygons of the same number of sides are similar, if
(i) their corresponding angles are equal and
(ii) their corresponding sides are in proportion or their corresponding sides are in the same ratio.
The same ratio of the corresponding sides is referred to as the representative fraction or the scale factor for the polygons.
Similar Triangles :
Two triangles are said to be similar,
if (i) their corresponding angles are equal and
ii) their corresponding sides are in proportion (are in the same ratio).
Basic Proportionality Theorem (or Thales Theorem) : 1
If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio. Or If a line is drawn parallel to one side of a triangle, intersecting the other two sides in distinct points, the other two sides are divided in the same ratio i .e.. If in ∆ABC, l∥ BC, intersecting in D and E. then
Converse of Basic Proportionality Theorem :
If a line divides any two sides of a triangle in the sameratio, the line is parallel to the third side i.e.
In ∆ABC, if l intersects AB in D and AC in E, such that:
Criteria for Similarity of Triangles:
Two triangles are said to be similar, if
(i) their corresponding angles are equal and (ii) their corresponding sides are in proportion (or are in the same ratio).
2 (i) AA or AAA Similarity Criterion : If two angles of one triangle are equal to two corresponding angles of another triangle, then the triangles are similar. If two angles of one triangle are respectively equal to the two angles of another triangle, then the third angles of the two triangles are necessarily equal, because the sum of three angles of a triangle is always 180 0 .
(ii) SAS Similarity Criterion : If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio, then the two triangles are similar. Or If two sides of a triangle are proportional to two corresponding sides of another triangle and the angles included between them are equal, then the triangles are similar.
iii) SSS Similarity Criterion : If in two triangles, sides of one triangle are proportional (or are in the same ratio) to the sides of the other triangle, then the triangles are similar. If ∆ABC~ ∆PQR by any one similarity criterion, then ∠A=∠P, ∠B=∠Q, ∠C=∠R and
i.e., A and P, B and Q, C and R are the corresponding vertices, also AB and PQ. BC and QR. CA and RP are the corresponding sides. 3 Areas of Similar Triangles: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. – The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians. – The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding altitudes. – The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding angle bisectors. Pythagoras Theorem : In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Converse of Pythagoras Theorem : In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and similar to each other i .e ..
If in ∆ABC, ∠B=90^0 and BD ⊥ AC, then (i) ∆ADB ~ ∆ABC (ii) ∆BDC ~ ∆ABC (iii) ∆ADB ~ ∆BDC
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