Class 9 Maths Chapter 6: Lines and Angles

Lines and Angles is Chapter 6 of NCERT Class 9 Maths. It revises the basic terms of geometry and then studies the angles formed when two lines intersect and when a transversal cuts two parallel lines, using axioms and simple proofs. These notes give you the key results, properties and solved examples you need for the CBSE exam.

What you will learn

Basic terms: line segment, ray, collinear points, angle and the types of angles
Complementary, supplementary and adjacent angles, and a linear pair
The Linear Pair Axiom and its converse
Why vertically opposite angles are equal
Angles formed by a transversal with parallel lines, and lines parallel to the same line

Key Concepts and Formulas

Basic Terms and Definitions

Geometry in this chapter uses a few simple building blocks. Make sure you can draw and name each one correctly.

Line segment AB: part of a line with two end points A and B
Ray AB: part of a line starting at A and going on endlessly through B
Collinear points: three or more points lying on the same line; otherwise they are non-collinear
Angle: formed when two rays start from the same point; the rays are the arms and the common point is the vertex

Types of Angles

Angles are classified by their measure.

Acute angle:    0° < θ < 90°
Right angle:    θ = 90°
Obtuse angle:   90° < θ < 180°
Straight angle: θ = 180°
Reflex angle:   180° < θ < 360°

Pairs of Angles

Some pairs of angles have special names because of their sum or their position.

Complementary angles: ∠A + ∠B = 90°
Supplementary angles: ∠A + ∠B = 180°
Linear pair:          ∠A + ∠B = 180°

Adjacent angles: have a common vertex and a common arm, and their other arms lie on opposite sides of the common arm
Linear pair: adjacent angles whose non-common arms are opposite rays
Vertically opposite angles: the pairs of opposite angles formed when two lines intersect

Linear Pair Axiom and Vertically Opposite Angles

These results are used in almost every question of this chapter. In the proof below, lines AB and CD intersect at O.

Axiom 6.1: ray stands on a line
  ⇒ sum of the two adjacent angles = 180°
Axiom 6.2: adjacent angles sum to 180°
  ⇒ non-common arms form a straight line
Theorem 6.1: vertically opposite angles are equal
Proof:
∠AOC + ∠AOD = 180° (linear pair)
∠AOD + ∠BOD = 180° (linear pair)
∴ ∠AOC = ∠BOD
Similarly ∠AOD = ∠BOC
Angles around a point: sum = 360°

Two intersecting lines forming vertically opposite angles of 72° and 108° and linear pairs, Class 9 Maths Lines and Angles — Vertically opposite angles are equal and linear pairs add to 180°

Parallel Lines and a Transversal

Two lines in a plane that never meet are parallel. A line that cuts two or more lines at distinct points is called a transversal. When a transversal cuts two lines, it forms eight angles, which are grouped into pairs. The results below hold when the two lines are parallel.

Corresponding angles: equal
Alternate interior angles: equal
Co-interior angles: sum = 180°

Parallel lines cut by a transversal showing corresponding, alternate interior and co-interior angles, Class 9 Maths Chapter 6 — Angle pairs formed when a transversal cuts parallel lines

Corresponding angles are in matching positions at the two intersections.
Alternate interior angles lie between the two lines, on opposite sides of the transversal.
Co-interior angles (interior angles on the same side) lie between the lines, on the same side of the transversal.
Converse results: if any pair of corresponding angles is equal, or alternate interior angles are equal, or co-interior angles add up to 180°, then the two lines are parallel.

Lines Parallel to the Same Line

This result helps you prove that two lines are parallel without measuring them: lines that are parallel to the same line are parallel to each other.

If l ∥ m and n ∥ m, then l ∥ n

Solved Examples

Example 1: Find the complement and the supplement of 38°.

Complement = 90° − 38° = 52°
Supplement = 180° − 38° = 142°
∴ Complement = 52°, supplement = 142°

Example 2: Two angles forming a linear pair are (3x + 10)° and (2x − 5)°. Find them.

(3x + 10) + (2x − 5) = 180
5x + 5 = 180
5x = 175
x = 35
3(35) + 10 = 115, 2(35) − 5 = 65
∴ The angles are 115° and 65°

Example 3: Two lines intersect and one of the angles formed is 72°. Find the other three angles.

Vertically opposite angle = 72°
Linear pair with 72° = 180° − 72° = 108°
Its vertically opposite angle = 108°
∴ The angles are 72°, 108° and 108°

Example 4: The supplement of an angle is five times its complement. Find the angle.

Let the angle be x
180 − x = 5(90 − x)
180 − x = 450 − 5x
4x = 270
∴ x = 67.5°

Example 5: Three angles on one side of a straight line at a point are 2x, 3x and 4x. Find them.

2x + 3x + 4x = 180°
9x = 180°
x = 20°
∴ The angles are 40°, 60° and 80°

Example 6: Two parallel lines are cut by a transversal and two corresponding angles are (5y − 20)° and (3y + 30)°. Find y and the angle.

Corresponding angles are equal
5y − 20 = 3y + 30
2y = 50
y = 25
Angle = 5 × 25 − 20 = 105
∴ y = 25 and each angle = 105°

Example 7: Two co-interior angles between parallel lines are in the ratio 2 : 3. Find them.

Let the angles be 2k and 3k
2k + 3k = 180°
5k = 180°
k = 36°
∴ The angles are 72° and 108°

Common Mistakes to Avoid

Do not treat every pair of adjacent angles as a linear pair; the non-common arms must form a straight line.
Corresponding and alternate angles are equal only when the lines are parallel; check this before using the property.
Co-interior angles are supplementary, not equal. Mixing this up is a very common error.
Always write the reason (for example "linear pair", "vertically opposite angles" or "alternate interior angles") next to each step in a proof question.

Quick Revision

Complementary: sum = 90°
Supplementary: sum = 180°
Linear pair: sum = 180°
Angles around a point: sum = 360°
Parallel lines: corresponding angles equal
Parallel lines: alternate interior angles equal
Parallel lines: co-interior angles sum = 180°
l ∥ m and n ∥ m ⇒ l ∥ n