Lines and Angles Chapter 6 Class 9 Maths Notes - Definitions, Theorems with Solved Examples

The chapter Lines and Angles class 9 maths notes is one of the basic and important topics in geometry for students of CBSE Board. These notes are specially prepared as per the latest CBSE syllabus of Class 9, so students can understand each concept step by step. In this chapter, you will learn about different types of lines, angles, pairs of angles.

These CBSE Class 9 Notes explain key ideas like intersecting lines, parallel lines, transversal, and angle properties using solved examples. The theorems of lines and angles class 9 are also covered clearly, so students can apply them in problems without confusion. Sometimes students feel this topic is tricky, but with proper CBSE notes it becomes much more easy to understand Concepts.

Students can acess lines and angles class 9 maths notes pdf for quick revision before class 9 maths exams, because it saves time and gives clear formulas and rules in one place. Overall, this chapter builds a strong base for higher maths concepts, even though at first it may look little confusing but it gets better with practice.

Core Concepts of Lines and Angles

Lines and Angles is one of the foundational chapters in Class 9 Mathematics. The concepts introduced here form the backbone of Euclidean geometry and are essential for understanding triangles, quadrilaterals, circles, and coordinate geometry in higher classes. Whether you're preparing for your board exams, competitive tests like NTSE or Olympiads, or simply building a strong mathematical base, a clear understanding of lines and angles is indispensable.

This comprehensive guide covers every concept from the chapter definitions, types of angles, properties of parallel lines cut by a transversal, important theorems with full proofs, and 20 carefully solved examples ranging from basic to advanced.

Basic Definitions

What is a Line?

A line is a one-dimensional geometric figure that has infinite length but no width and no thickness. A line extends endlessly in both directions and is represented by a straight mark with arrows on both ends (↔). A line segment is a part of a line bounded by two definite endpoints, while a ray is a part of a line that starts at one point and extends infinitely in one direction.

What is an Angle?

An angle is formed by the union of two non-collinear rays that share a common initial point called the vertex. The two rays forming the angle are called its arms. Angles are measured in degrees (°).

Angle Addition Axiom: If point X lies in the interior of ∠BAC, then:

m∠BAC = m∠BAX + m∠XAC

This axiom tells us that an angle can be split into smaller angles, and the parts always add up to the whole.

Important Measurement Units:

• 1 right angle = 90°
• 1° = 60′ (minutes)
• 1′ = 60″ (seconds)

Types of Angles

Understanding the classification of angles is the starting point of geometry. Each type has a specific range of measure and unique properties.

cute Angle

An angle whose measure is greater than 0° and less than 90° is called an acute angle.

Example: 30°, 45°, 60°, 75° are all acute angles.

Right Angle

An angle whose measure is exactly 90° is called a right angle. The two arms of a right angle are perpendicular to each other.

Example: The corner of a square or a rectangle.

Obtuse Angle

An angle whose measure is greater than 90° but less than 180° is called an obtuse angle.

Example: 120°, 135°, 150°.

Straight Angle

An angle whose measure is exactly 180° is called a straight angle. Its arms form a straight line.

Example: ∠AOB = 180° when A, O, B are collinear.

Reflex Angle

An angle whose measure is greater than 180° but less than 360° is called a reflex angle.

Example: 270°, 300°, 330°.

Complete Angle

An angle whose measure is exactly 360° is called a complete angle or a full rotation.

Pairs of Angles

Complementary Angles

Two angles are complementary if the sum of their measures equals 90°.

Each angle is called the complement of the other.

Example: 30° and 60° are complementary (30° + 60° = 90°).

Note: If one angle is x°, its complement is (90 – x)°.

Supplementary Angles

Two angles are supplementary if the sum of their measures equals 180°.

Example: 120° and 60° are supplementary (120° + 60° = 180°).

Note: If one angle is x°, its supplement is (180 – x)°.

Adjacent Angles

Two angles are adjacent if they satisfy all three conditions:

• They share the same vertex
• They share a common arm
• Their non-common arms lie on opposite sides of the common arm

Linear Pair of Angles

Two adjacent angles form a linear pair if their non-common arms are opposite rays (i.e., they together form a straight line).

The sum of angles in a linear pair is always 180°.

Linear Pair Axiom: If a ray stands on a line, the two adjacent angles so formed are supplementary.

Vertically Opposite Angles

When two lines intersect at a point, they form two pairs of vertically opposite angles. Vertically opposite angles are always equal.

Example: If lines AB and CD intersect at O:

• ∠AOC = ∠BOD
• ∠AOD = ∠BOC

Angle Bisector

A ray OX is said to be the bisector of ∠AOB if X lies in the interior of ∠AOB and ∠AOX = ∠BOX. The bisector divides the angle into two equal halves.

Angles Made by a Transversal with Parallel Lines

This is the most tested topic from this chapter in board examinations.

What is a Transversal?

A transversal is a line that intersects two or more lines (usually parallel) at distinct points. When a transversal crosses two parallel lines, it creates 8 angles in total 4 at each intersection point.

Corresponding Angles

Corresponding angles are on the same side of the transversal and both lie either above or below the parallel lines.

Pairs: ∠1 & ∠5, ∠2 & ∠6, ∠3 & ∠7, ∠4 & ∠8

Corresponding Angles Axiom: When a transversal intersects two parallel lines, each pair of corresponding angles is equal.

Conversely, if corresponding angles are equal, the lines are parallel.

Alternate Interior Angles

These lie on opposite sides of the transversal and between (interior to) the two parallel lines.

Pairs: ∠3 & ∠5, ∠2 & ∠8 (using standard numbering)

Theorem: Alternate interior angles are equal when lines are parallel.

Alternate Exterior Angles

These lie on opposite sides of the transversal and outside the parallel lines.

Pairs: ∠1 & ∠7, ∠2 & ∠8

Alternate exterior angles are also equal when lines are parallel.

Consecutive Interior Angles (Co-interior / Same-side Interior Angles)

These lie on the same side of the transversal and between the parallel lines.

Pairs: ∠2 & ∠5, ∠3 & ∠8

Theorem: Consecutive interior angles are supplementary (add up to 180°) when lines are parallel.

Important Theorems with Proofs

Theorem 1: Vertically Opposite Angles are Equal

Statement: If two lines intersect each other, then the vertically opposite angles are equal.

Given: Lines AB and CD intersect at point O.

To Prove:

• (i) ∠AOC = ∠BOD
• (ii) ∠BOC = ∠AOD

Proof:

Since ray OD stands on line AB:

∠AOD + ∠DOB = 180° …(i) [Linear pair]

Since ray OA stands on line CD:

∠AOC + ∠AOD = 180° …(ii) [Linear pair]

From equations (i) and (ii):

∠AOD + ∠DOB = ∠AOC + ∠AOD

∴ ∠DOB = ∠AOC

∴ ∠AOC = ∠BOD

Similarly, ∠BOC = ∠AOD can be proved.

Hence Proved.

Theorem 2: Alternate Interior Angles are Equal (for Parallel Lines)

Statement: If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.

Given: AB ∥ CD; transversal l intersects AB at P and CD at Q, forming alternate interior angle pairs ∠1, ∠2 and ∠3, ∠4.

To Prove: ∠1 = ∠2 and ∠3 = ∠4

Proof:

∠2 = ∠5 [Vertically opposite angles]

∠1 = ∠5 [Corresponding angles, since AB ∥ CD]

∴ ∠1 = ∠2

∠3 = ∠6 [Vertically opposite angles]

∠4 = ∠6 [Corresponding angles, since AB ∥ CD]

∴ ∠3 = ∠4

Hence Proved.

Theorem 3: Lines Parallel to the Same Line are Parallel to Each Other

Statement: If two lines are each parallel to a third line, then they are parallel to each other.

This theorem follows from the transitive property of parallel lines and is used frequently in multi-line geometry problems.

Lines and Angles Important Facts and Properties to Remember

1. If a ray stands on a line, the sum of adjacent angles formed = 180°.
2. If the sum of two adjacent angles is 180°, their non-common arms are opposite rays (they form a straight line).
3. The sum of all angles around a point = 360°.
4. Vertically opposite angles are always equal.
5. When a transversal cuts two parallel lines: corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.
6. If a transversal makes a pair of equal alternate interior angles with two lines, those lines are parallel.
7. If a transversal makes a pair of supplementary consecutive interior angles with two lines, those lines are parallel.
8. If two parallel lines are cut by a transversal, the bisectors of alternate interior angles are parallel.
9. If two parallel lines are cut by a transversal, the bisectors of corresponding angles are also parallel.
10. A line perpendicular to one of two parallel lines is also perpendicular to the other.
11. Two angles with parallel arms are either equal or supplementary.
12. Two angles with perpendicular arms are either equal or supplementary.

Lines and Angles Class 9 Maths Solved Examples  

Q: Two supplementary angles are in the ratio 4 : 5. Find the angles.

Solution:

Let the angles be 4x and 5x.

Since they are supplementary: 4x + 5x = 180°

9x = 180° → x = 20°

∴ Angles are 4 × 20° = 80° and 5 × 20° = 100°

Q: An angle differs from its complement by 10°. Find the angle.

Solution:

Let the angle = x°. Its complement = (90 – x)°

Given: x – (90 – x) = 10

x – 90 + x = 10

2x = 100 → x = 50°

∴ The required angle is 50°

Q: Lines AB, CD, and EF intersect at O. If ∠AOE = 40° and ∠BOD = 35°, find ∠AOC, ∠DOE, and ∠BOF.

Solution:

∠AOC = ∠BOD = 35° [Vertically opposite]

∠BOF = ∠AOE = 40° [Vertically opposite]

∠AOB = 180° [Straight angle]

∠AOC + ∠COF + ∠BOF = 180°

35° + ∠COF + 40° = 180°

∠COF = 105°

∠DOE = ∠COF = 105° [Vertically opposite]

∴ ∠AOC = 35°, ∠DOE = 105°, ∠BOF = 40°

Q: If l ∥ m, n ∥ p and ∠1 = 85°, find ∠2.

Solution:

Since n ∥ p and m is a transversal:

∠1 = ∠3 = 85° [Corresponding angles]

Since m ∥ l and p is a transversal:

∠2 + ∠3 = 180° [Consecutive interior angles]

∠2 + 85° = 180°

∴ ∠2 = 95°

Q: OP and OQ bisect ∠BOC and ∠AOC respectively. Prove that ∠POQ = 90°.

Solution:

OP bisects ∠BOC → ∠POC = ½ ∠BOC …(i)

OQ bisects ∠AOC → ∠COQ = ½ ∠AOC …(ii)

OC stands on AB → ∠AOC + ∠BOC = 180° [Linear pair]

½ ∠AOC + ½ ∠BOC = 90°

∠COQ + ∠POC = 90° [From (i) and (ii)]

∴ ∠POQ = 90° [By angle sum property]

Hence Proved.

Q: Find the complement of 37°.

Solution:

Complement = 90° – 37° = 53°

Q: Find the supplement of 112°.

Solution:

Supplement = 180° – 112° = 68°

Q: Two lines AB and CD intersect at O. If ∠AOC = 50°, find all four angles.

Solution:

∠AOC = 50° (given)

∠BOD = ∠AOC = 50° [Vertically opposite]

∠AOD = 180° – 50° = 130° [Linear pair]

∠BOC = ∠AOD = 130° [Vertically opposite]

∴ The four angles are 50°, 130°, 50°, 130°

Q: An angle is 4 times its complement. Find the angle.

Solution:

Let the angle = x°. Complement = (90 – x)°

x = 4(90 – x)

x = 360 – 4x

5x = 360 → x = 72°

∴ The angle is 72°

Q: An angle is one-third of its supplement. Find the angle.

Solution:

Let the angle = x°. Supplement = (180 – x)°

x = (180 – x)/3

3x = 180 – x

4x = 180 → x = 45°

∴ The angle is 45°

Q: In a linear pair, if one angle is 25° more than the other, find both angles.

Solution:

Let the smaller angle = x°. The other = (x + 25)°

x + x + 25 = 180

2x = 155 → x = 77.5°

∴ Angles are 77.5° and 102.5°

Q: Two complementary angles are in the ratio 2 : 3. Find them.

Solution:

Let angles be 2x and 3x.

2x + 3x = 90° → 5x = 90° → x = 18°

∴ Angles are 36° and 54°

Q: If two parallel lines are cut by a transversal and one of the co-interior angles is 65°, find the other.

Solution:

Co-interior (consecutive interior) angles are supplementary.

Other angle = 180° – 65° = 115°

Q: A transversal cuts two parallel lines. One alternate interior angle is (3x + 10)° and the other is (5x – 30)°. Find x and the angles.

Solution:

Alternate interior angles are equal:

3x + 10 = 5x – 30

40 = 2x → x = 20

Each angle = 3(20) + 10 = 70°

Q: Find the value of x if corresponding angles formed by a transversal with two parallel lines are (2x + 40)° and (3x – 10)°.

Solution:

Corresponding angles are equal:

2x + 40 = 3x – 10

50 = x

∴ x = 50; Each angle = 140°

Q: The angles of a linear pair are (5y – 20)° and (3y + 8)°. Find y.

Solution:

(5y – 20) + (3y + 8) = 180

8y – 12 = 180

8y = 192 → y = 24

∴ Angles are 100° and 80°

Q: Two lines intersect. One of the angles formed is a right angle. Show that all four angles are right angles.

Solution:

Let ∠AOC = 90°.

∠AOD = 180° – 90° = 90° [Linear pair]

∠BOD = ∠AOC = 90° [Vertically opposite]

∠BOC = ∠AOD = 90° [Vertically opposite]

∴ All four angles are 90° — the lines are perpendicular. Proved.

Q: The sum of two angles is 200° and their difference is 20°. Find the angles.

Solution:

Let angles be x and y where x > y.

x + y = 200°, x – y = 20°

Adding: 2x = 220° → x = 110°

y = 200° – 110° = 90°

∴ Angles are 110° and 90°

Q: Ray OC stands on line AB. If ∠AOC : ∠BOC = 2 : 3, find the angles.

Solution:

∠AOC + ∠BOC = 180° [Linear pair]

Let ∠AOC = 2k and ∠BOC = 3k

2k + 3k = 180° → k = 36°

∴ ∠AOC = 72° and ∠BOC = 108°

Q: In the figure, if AB ∥ CD and ∠APQ = 50°, ∠PRD = 127°, find ∠QPR.

Solution:

Since AB ∥ CD and PQ is a transversal:

∠PQR = ∠APQ = 50° [Alternate interior angles — drawing a line through R parallel to AB]

∠PRQ = 180° – 127° = 53° [Linear pair at R with CD]

In △PQR:

∠QPR = 180° – 50° – 53° = 77°

Lines and Angles Quick Revision

Class 9 Maths Lines and Angles Formulas

Applications of Lines and Angles in Real Life

• Architecture and Engineering: Architects use angle properties to design structurally sound buildings. Parallel walls, perpendicular floors, and transversal beams are all based on lines-and-angles principles.
• Road Design: Highway designers use alternate angles and parallel lines to create smooth lane transitions and ensure equal road widths.
• Navigation: Pilots and sailors use angle concepts to chart courses using bearings and transversal-like paths across maps.
• Art and Design: Perspective drawing relies on parallel lines converging at vanishing points — a direct application of transversal and corresponding angle concepts.
• Computer Graphics: Every pixel arrangement on a screen and every rotational transformation in graphics software is governed by angle mathematics.