📐 Lines and Angles - Class 9

Complete notes with theorems, proofs, and formula sheet

1. Introduction to Lines and Angles

Lines and angles are fundamental concepts in geometry. Understanding their properties and relationships is essential for solving geometric problems and proving theorems.

📖 Basic Concepts

Point: A location in space with no dimensions (no length, width, or height).

Line: Extends infinitely in both directions, has no endpoints.

Line Segment: Part of a line with two endpoints.

Ray: Starts from a point and extends infinitely in one direction.

Angle: Formed by two rays sharing a common endpoint (vertex).

2. Types of Angles

2.1 Based on Measure

Type	Measure	Description
Acute Angle	0° < θ < 90°	Less than a right angle
Right Angle	θ = 90°	Exactly 90 degrees
Obtuse Angle	90° < θ < 180°	Greater than right, less than straight
Straight Angle	θ = 180°	Forms a straight line
Reflex Angle	180° < θ < 360°	Greater than straight angle
Complete Angle	θ = 360°	Full rotation

2.2 Complementary and Supplementary Angles

Complementary Angles: Two angles whose sum is 90°.
Example: 30° and 60° are complementary (30° + 60° = 90°).
Supplementary Angles: Two angles whose sum is 180°.
Example: 110° and 70° are supplementary (110° + 70° = 180°).
Complementary angles don't need to be adjacent.
Supplementary angles don't need to be adjacent either.

📝 Example: Finding Angles

Q1: If two angles are complementary and one is 35°, find the other.

Solution:

Sum of complementary angles = 90°

Let other angle = x

35° + x = 90°

x = 90° - 35° = 55°

Answer: 55°

Q2: Find the supplement of 125°.

Solution:

Supplement = 180° - 125° = 55°

Answer: 55°

2.3 Adjacent Angles

Two angles are adjacent if they share a common vertex and a common arm.
The other arms are on opposite sides of the common arm.
Adjacent angles don't overlap.
Example: If two angles share vertex O and arm OA, they're adjacent if on different sides of OA.

2.4 Linear Pair

A linear pair consists of two adjacent angles whose sum is 180°.
The non-common arms form a straight line.
Linear Pair Axiom: If two angles form a linear pair, their sum is 180°.
All linear pairs are supplementary, but not all supplementary angles form a linear pair.

2.5 Vertically Opposite Angles

When two lines intersect, they form four angles.
The angles opposite to each other are called vertically opposite angles.
Theorem: Vertically opposite angles are always equal.
If lines AB and CD intersect at O, then ∠AOC = ∠BOD and ∠AOD = ∠BOC.

📝 Example: Vertically Opposite Angles

Q: Two lines intersect at a point. If one angle is 65°, find all four angles.

Solution:

Let the angle be ∠1 = 65°

∠3 = ∠1 = 65° (vertically opposite)

∠1 + ∠2 = 180° (linear pair)

∠2 = 180° - 65° = 115°

∠4 = ∠2 = 115° (vertically opposite)

Angles: 65°, 115°, 65°, 115°

3. Parallel Lines and Transversal

📖 Definitions

Parallel Lines: Lines in the same plane that never intersect, no matter how far extended.

Symbol: ∥ (AB ∥ CD means line AB is parallel to line CD)

Transversal: A line that intersects two or more lines at distinct points.

3.1 Angles Formed by a Transversal

When a transversal intersects two lines, it creates 8 angles. These angles have special names and relationships:

Corresponding Angles: Angles in the same relative position at each intersection.
Alternate Interior Angles: Interior angles on opposite sides of the transversal.
Alternate Exterior Angles: Exterior angles on opposite sides of the transversal.
Co-interior Angles (Consecutive Interior): Interior angles on the same side of transversal.

3.2 Properties When Lines are Parallel

⚠️ Important Theorems

When a transversal intersects two parallel lines:

1. Corresponding angles are equal.

2. Alternate interior angles are equal.

3. Alternate exterior angles are equal.

4. Co-interior angles are supplementary (sum = 180°).

📝 Example: Parallel Lines Problems

Q: Two parallel lines are cut by a transversal. If one corresponding angle is 70°, find all angles.

Solution:

Let one angle = 70°

Corresponding angle = 70° (corresponding angles equal for parallel lines)

Alternate interior angle = 70°

Vertically opposite = 70°

Linear pairs with 70° = 180° - 70° = 110°

Angles: Four 70° and four 110° angles

3.3 Tests for Parallel Lines

Two lines are parallel if any of these conditions hold when cut by a transversal:

Corresponding angles are equal.
Alternate interior angles are equal.
Alternate exterior angles are equal.
Co-interior angles are supplementary.

4. Angle Sum Property of a Triangle

⚠️ Important Theorem

Angle Sum Property: The sum of the three angles of a triangle is 180°.

If a triangle has angles A, B, and C, then: A + B + C = 180°

This is one of the most fundamental theorems in geometry!

4.1 Proof of Angle Sum Property

Draw a triangle ABC with angles ∠A, ∠B, and ∠C.
Draw a line through A parallel to BC.
Use alternate interior angles theorem.
Show that angles at A form a straight line (180°).
Conclude that ∠A + ∠B + ∠C = 180°.

📝 Example: Triangle Angles

Q: In triangle ABC, ∠A = 50° and ∠B = 60°. Find ∠C.

Solution:

Using angle sum property:

∠A + ∠B + ∠C = 180°

50° + 60° + ∠C = 180°

∠C = 180° - 110°

∠C = 70°

4.2 Exterior Angle Property

⚠️ Exterior Angle Theorem

An exterior angle of a triangle equals the sum of the two opposite interior angles.

If ∠ACD is an exterior angle at C, then: ∠ACD = ∠A + ∠B

An exterior angle is formed when one side is extended.
Each vertex has two exterior angles (one on each side).
Exterior angle + adjacent interior angle = 180° (linear pair).
The exterior angle is always greater than either non-adjacent interior angle.

📝 Example: Exterior Angle

Q: In triangle PQR, ∠P = 40°, ∠Q = 65°. Find the exterior angle at R.

Solution:

Exterior angle at R = ∠P + ∠Q (exterior angle property)

= 40° + 65°

= 105°

Verification:

∠R = 180° - (40° + 65°) = 75°

Exterior at R = 180° - 75° = 105° ✓

📚 Quick Formula Sheet - Lines and Angles

Types of Angles

Acute: 0° < θ < 90°

Right: θ = 90°

Obtuse: 90° < θ < 180°

Straight: θ = 180°

Reflex: 180° < θ < 360°

Angle Pairs

Complementary: Sum = 90°

Supplementary: Sum = 180°

Linear Pair: Adjacent + Sum = 180°

Vertically Opposite: Equal

Parallel Lines (∥)

Corresponding angles = Equal

Alternate interior = Equal

Alternate exterior = Equal

Co-interior angles = 180°

When cut by transversal

Triangle Angle Sum

∠A + ∠B + ∠C = 180°

Sum of three angles

Exterior Angle:

= Sum of opposite interior angles

Important Results

Angles on a straight line = 180°

Angles around a point = 360°

Exterior angle > Each non-adjacent interior angle

Fundamental properties

Finding Unknown Angles

Complement of x = 90° - x

Supplement of x = 180° - x

Vertically opposite = Same

Linear pair = 180° - angle

Parallel Line Tests

If corresponding angles equal → Lines ∥

If alternate interior equal → Lines ∥

If co-interior = 180° → Lines ∥

Converse theorems

Quick Tips

• Always mark known angles

• Look for angle relationships

• Use properties systematically

• Verify your answer

Problem-solving approach

💡 Study Tips for Lines and Angles

• Draw clear diagrams - half the solution is in the diagram!

• Mark all known angles and parallel lines on your diagram.

• Learn to identify angle relationships quickly (vertically opposite, linear pair, etc.).

• Practice identifying corresponding, alternate, and co-interior angles.

• Remember: Parallel lines have special properties with transversals.

• The angle sum property (180°) is crucial for triangles.

• Exterior angle = sum of two opposite interior angles - memorize this!

• Always verify your answer makes sense (angles should be positive and less than 360°).

🔑 Common Mistakes to Avoid

Don't confuse corresponding with alternate angles.
Remember: Vertically opposite angles are EQUAL, not supplementary.
Co-interior angles sum to 180° (not equal) for parallel lines.
Linear pair must be adjacent - not all supplementary angles are linear pairs.
Exterior angle equals sum of TWO opposite interior angles, not one.
Draw accurate diagrams - wrong diagrams lead to wrong answers!
Don't assume lines are parallel unless stated or proved.