Lines and Angles

Board: GSEB | Class: Std 9

Comprehensive study notes for Lines and Angles by Ajay Yadav (Math King of Katargam). Master every concept with clear explanations, solved examples, and practice problems.

Key Concepts

Basic Terms

A line segment has two endpoints. A ray has one endpoint. A line extends infinitely in both directions. An angle is formed when two rays originate from a common point (vertex).

Types of Angles

∠ can be: Acute (0°<θ<90°), Right (θ=90°), Obtuse (90°<θ<180°), Straight (θ=180°), Reflex (180°<θ<360°), Complete (θ=360°).

Complementary & Supplementary

Complementary angles: Sum = 90°. Supplementary angles: Sum = 180°.

Adjacent & Linear Pair

Adjacent angles share a common arm and vertex. A linear pair is formed when two adjacent angles sum to 180°.

Vertically Opposite Angles

When two lines intersect, vertically opposite angles are equal. If lines AB and CD intersect at O, then ∠AOC = ∠BOD and ∠AOD = ∠BOC.

Parallel Lines & Transversal

When a transversal cuts two parallel lines: Corresponding angles are equal, Alternate interior angles are equal, Co-interior angles sum to 180°.

Angle Sum of a Triangle

The sum of the three interior angles of a triangle is 180°. If a side is extended, the exterior angle equals the sum of the two opposite interior angles.

Important Formulas

Linear Pair	`∠1 + ∠2 = 180°`
Vertically Opposite	`∠AOC = ∠BOD`
Corresponding Angles	`∠1 = ∠5, ∠2 = ∠6, ∠3 = ∠7, ∠4 = ∠8`
Alternate Interior	`∠3 = ∠6, ∠4 = ∠5`
Co-interior (Consecutive)	`∠3 + ∠5 = 180°, ∠4 + ∠6 = 180°`
Angle Sum of Triangle	`∠A + ∠B + ∠C = 180°`
Exterior Angle	`Exterior ∠ = Sum of opposite interior angles`

Solved Examples

Example 1: Find the angle which is equal to its complement.

Solution: Let the angle be x. Then x + x = 90° ⇒ 2x = 90° ⇒ x = 45°

Example 2: If two adjacent angles on a straight line are (3x-10)° and (2x+20)°, find x.

Solution: Since they form a linear pair: (3x-10) + (2x+20) = 180 ⇒ 5x + 10 = 180 ⇒ 5x = 170 ⇒ x = 34

Example 3: In ΔABC, ∠A = 60° and ∠B = 70°. Find ∠C.

Solution: ∠A + ∠B + ∠C = 180° ⇒ 60 + 70 + ∠C = 180 ⇒ ∠C = 50°

Practice Questions

Find the supplement of 125°.
If two lines intersect, prove that vertically opposite angles are equal.
Two parallel lines are cut by a transversal. If one interior angle is 110°, find all other angles.
In ΔPQR, ∠P = 2∠Q and ∠R = 3∠Q. Find all angles.
An exterior angle of a triangle is 120° and its interior opposite angles are equal. Find each angle.

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