1. Introduction

This chapter explores the properties of lines in geometry:

Parallel lines – lines that never meet, no matter how far they are extended.
Intersecting lines – lines that cross each other at a point.
Transversals – a line that cuts across two or more lines.

These concepts help in understanding angles formed by lines, which is essential in geometry.

2. Intersecting Lines

Two lines that meet at a point are called intersecting lines.
The point of intersection is where the lines cross.
Angles formed at the intersection:
1. Vertically opposite angles – always equal.
2. Adjacent angles – add up to 180° (linear pair).

Example:
If two lines AB and CD intersect at O:

∠AOC = ∠BOD (vertically opposite)
∠AOB + ∠BOC = 180°

3. Parallel Lines

Definition: Two lines in a plane that never meet are called parallel lines.
Notation: AB ∥ CD

Key Properties:

When a transversal cuts parallel lines, it forms specific types of angles:
1. Corresponding angles (∠1 = ∠2)
2. Alternate interior angles (∠3 = ∠4)
3. Alternate exterior angles (∠5 = ∠6)
4. Co-interior angles (∠7 + ∠8 = 180°)

4. Types of Angles Formed by a Transversal

Corresponding angles: Same relative position; equal if lines are parallel.
Alternate interior angles: On opposite sides of transversal, inside lines; equal if lines are parallel.
Alternate exterior angles: On opposite sides of transversal, outside lines; equal if lines are parallel.
Co-interior (Consecutive interior) angles: Inside lines, same side of transversal; sum = 180° if lines are parallel.

Example Diagram:

AB ∥ CD, transversal EF
∠1 = ∠5 (corresponding)
∠3 = ∠6 (alternate interior)
∠2 + ∠4 = 180° (co-interior)

5. Properties of Parallel Lines and Transversals

Property	Condition/Formula
Corresponding angles	∠1 = ∠2
Alternate interior angles	∠3 = ∠4
Alternate exterior angles	∠5 = ∠6
Co-interior angles	∠7 + ∠8 = 180°

Note: These properties are true only if lines are parallel.

6. Special Cases

Intersecting lines: Vertically opposite angles are equal.
Transversal with non-parallel lines: Angles do not follow above rules.
Parallel lines proof: Equal corresponding angles or sum of co-interior angles = 180° can prove lines are parallel.

7. Examples

Two lines AB ∥ CD, transversal EF cuts them. If one of the corresponding angles is 70°, find all other angles formed.
- Corresponding angles = 70°
- Alternate interior angles = 70°
- Co-interior angles = 110° (180 – 70)
Two intersecting lines form an angle of 40°. Find all remaining angles.
- Vertically opposite angles = 40°
- Adjacent angles = 140°

8. Summary Table

Concept	Key Points
Intersecting lines	Meet at a point; vertically opposite angles equal
Parallel lines	Never meet; equal corresponding/alternate angles
Transversal	Line cutting two or more lines
Corresponding angles	Same position; equal if lines parallel
Alternate interior angles	Opposite sides inside; equal if lines parallel
Alternate exterior angles	Opposite sides outside; equal if lines parallel
Co-interior angles	Same side inside; sum = 180°

9. Tips for the Chapter

Always label angles clearly when drawing diagrams.
Remember the four main types of angles formed by a transversal.
Use vertically opposite angles and linear pairs to find unknown angles.
Check if lines are parallel using corresponding/alternate/co-interior angles.

50 Questions

Section A: Multiple Choice Questions (MCQs) – 10 Questions

Two lines that meet at a point are called:
a) Parallel lines
b) Intersecting lines
c) Transversals
d) Perpendicular lines
Lines that never meet are called:
a) Intersecting lines
b) Perpendicular lines
c) Parallel lines
d) Transversals
A line cutting two or more lines is called:
a) Intersecting line
b) Parallel line
c) Transversal
d) Perpendicular line
Vertically opposite angles are:
a) Equal
b) Supplementary
c) Complementary
d) Adjacent
When a transversal cuts two parallel lines, the corresponding angles are:
a) Equal
b) Sum = 180°
c) Sum = 90°
d) None of these
When a transversal cuts two parallel lines, co-interior angles:
a) Are equal
b) Sum = 180°
c) Sum = 90°
d) Are complementary
Alternate interior angles are:
a) Inside lines, same side
b) Inside lines, opposite sides
c) Outside lines, opposite sides
d) Outside lines, same side
Two intersecting lines form one angle of 70°. What is the vertically opposite angle?
a) 70°
b) 110°
c) 140°
d) 35°
If AB ∥ CD and a transversal cuts them, one alternate interior angle = 50°, then the other is:
a) 50°
b) 130°
c) 100°
d) 40°
Sum of angles on the same side of a transversal cutting two parallel lines is:
a) 90°
b) 180°
c) 360°
d) 120°

Section B: Fill in the Blanks – 10 Questions

Two lines that never meet are called _______.
A line that intersects two or more lines is called _______.
Vertically opposite angles are always _______.
Co-interior angles add up to _______ degrees.
Alternate interior angles lie _______ the parallel lines.
Corresponding angles lie _______ relative to the parallel lines.
Two intersecting lines form _______ pairs of vertically opposite angles.
If one angle of a linear pair is 110°, the other angle is _______.
A transversal makes _______ pairs of corresponding angles with two parallel lines.
Sum of all angles at the point of intersection of two lines is _______ degrees.

Section C: Short Answer Questions – 10 Questions

Draw two intersecting lines and label the vertically opposite angles.
Two lines intersect to form one angle of 40°. Find all remaining angles.
Draw two parallel lines cut by a transversal and label all types of angles.
Find the value of x if corresponding angles of parallel lines are x and 70°.
If one co-interior angle = 120°, find the other.
Find the missing alternate interior angle if one angle = 65°.
Two intersecting lines form angles in the ratio 2:3. Find all angles.
Draw a diagram to show alternate exterior angles and label them.
Name all types of angles formed when a transversal cuts two parallel lines.
If one corresponding angle = 3x + 10°, the other = 70°, find x.

Section D: Long Answer / Problem-Solving – 10 Questions

Two parallel lines are cut by a transversal. One angle = 80°. Find all other angles.
Draw two intersecting lines and measure all vertically opposite angles.
A transversal cuts two parallel lines forming angles 5x, 3x. Find x.
Two lines intersect at a point. One angle = 120°. Find the remaining angles.
Co-interior angles are in ratio 2:3. Find their measures.
Alternate interior angles are in ratio 4:5. Find their measures.
Draw two parallel lines and a transversal. Mark corresponding, alternate interior, alternate exterior, and co-interior angles.
One angle of a linear pair is twice the other. Find both angles.
The sum of alternate interior and co-interior angles = ? Discuss with a diagram.
A transversal intersects two parallel lines. If one exterior angle = 110°, find all remaining angles.

Section E: Application / Higher-Order Thinking – 10 Questions

A bridge’s supports are parallel. A diagonal beam crosses them. Find corresponding and alternate angles.
A flagpole casts shadows forming angles with two parallel roads. Find co-interior angles.
Two railway tracks are parallel. A transversal represents a crossing path. Label all types of angles.
Design a diagram with two intersecting lines and calculate vertically opposite angles if one angle = 50°.
Two parallel lines are cut by a transversal. One angle = 3x – 10°, another = 50°. Find x.
Verify the property: sum of co-interior angles = 180° using a diagram.
Draw two parallel lines cut by a transversal. If alternate interior angle = 70°, find corresponding and co-interior angles.
If two lines intersect at point O, one angle = 2y + 10°, the vertically opposite angle = 70°. Find y.
Two parallel lines are cut by a transversal. Prove that alternate exterior angles are equal.
A transversal intersects two parallel streets. One angle = 4x + 20°, its corresponding angle = 80°. Find x.

Answers

Section A: MCQs – Answers

b) Intersecting lines – Lines that meet at a point.
c) Parallel lines – Lines that never meet.
c) Transversal – A line that cuts two or more lines.
a) Equal – Vertically opposite angles are always equal.
a) Equal – Corresponding angles are equal when lines are parallel.
b) Sum = 180° – Co-interior angles are supplementary.
b) Inside lines, opposite sides – Alternate interior angles definition.
a) 70° – Vertically opposite angles are equal.
a) 50° – Alternate interior angles are equal for parallel lines.
b) 180° – Sum of co-interior angles = 180°.

Section B: Fill in the Blanks – Answers

Parallel lines
Transversal
Equal
180°
Between (the parallel lines)
Same relative position
Two pairs
70° – 180 – 110 = 70°
Four pairs
360° – sum of angles at a point = 360°

Section C: Short Answer Questions – Answers

Draw two intersecting lines AB and CD; vertically opposite angles: ∠AOC = ∠BOD, ∠AOD = ∠BOC.
Intersecting lines forming 40°: Vertically opposite angle = 40°, adjacent angles = 140°.
Two parallel lines AB ∥ CD cut by EF; label:

Corresponding: ∠1 = ∠5, ∠2 = ∠6
Alternate interior: ∠3 = ∠6, ∠4 = ∠5
Co-interior: ∠3 + ∠5 = 180°, ∠4 + ∠6 = 180°

Corresponding angles: x = 70° → x = 70° (direct equality)
Co-interior angles sum = 180° → other angle = 180 – 120 = 60°
Alternate interior angles equal → other angle = 65°
Intersecting lines angles in ratio 2:3 → 2x + 3x = 180 → 5x = 180 → x = 36° → angles: 72°, 108°, 72°, 108°
Draw two parallel lines and a transversal; alternate exterior angles on opposite sides = equal
Types of angles: Corresponding, Alternate Interior, Alternate Exterior, Co-interior, Vertically Opposite, Adjacent (Linear Pair)
Corresponding angles: 3x + 10 = 70 → 3x = 60 → x = 20

Section D: Long Answer / Problem-Solving – Answers

One angle = 80° → Corresponding = 80°, Alternate interior = 80°, Co-interior = 100°, Vertically opposite angles = equal
Draw intersecting lines; measure vertically opposite angles – they are equal
5x = 5x → x = ?; Solve accordingly if angles = 5x and 3x → 5x + 3x = 180 → 8x = 180 → x = 22.5°
Intersecting lines, one angle = 120° → vertically opposite = 120°, adjacent = 60° each
Co-interior angles ratio 2:3 → sum = 180° → 2x + 3x = 180 → x = 36° → angles = 72°, 108°
Alternate interior angles ratio 4:5 → 4x + 5x = 180 → x = 20° → angles = 80°, 100°
Diagram shows all angles formed by transversal cutting parallel lines; mark corresponding, alternate interior/exterior, co-interior
Linear pair: angles in ratio 2:1 → sum = 180 → 2x + x = 180 → 3x = 180 → x = 60° → angles = 120° and 60°
Diagram: Sum of co-interior angles = 180°; verify with measurements
Exterior angle = 110° → alternate exterior = 110°, co-interior angles = 70° each, vertically opposite = 110°, corresponding = 110°

Section E: Application / Higher-Order Thinking – Answers

Bridge supports: label angles formed by diagonal beam → corresponding, alternate interior/exterior, co-interior angles using parallel lines
Flagpole shadows: measure angles between shadow and roads → co-interior angles = 180°
Railway tracks: transversal = crossing path → label all angles
Intersecting lines with one angle = 50° → vertically opposite = 50°, adjacent = 130°
Corresponding angles: 3x – 10 = 50 → 3x = 60 → x = 20°
Co-interior angles sum = 180°; diagram shows verification
Parallel lines, transversal: alternate interior = 70°, co-interior = 110°, corresponding = 70°, alternate exterior = 70°
Intersecting lines: 2y + 10 = 70 → 2y = 60 → y = 30°
Prove alternate exterior angles = equal → use parallel line property and diagram
Corresponding angles: 4x + 20 = 80 → 4x = 60 → x = 15°