Lines and Angles: Practice Questions & Notes

Line

A line is a straight one-dimensional figure having no thickness and extending infinitely in both directions.

A line segment is a part of a line with two endpoints.
A ray is a line with one endpoint and extends infinitely in the other direction.

Angle

An angle is formed when two rays (or lines) meet at a common endpoint called the vertex.

Types of Angles

Angle Type	Measure
Acute Angle	Between 0° and 90°
Right Angle	Exactly 90°
Obtuse Angle	Between 90° and 180°
Straight Angle	Exactly 180°
Reflex Angle	Between 180° and 360°
Full Angle	Exactly 360°

Angle Pairs and Relationships

1. Complementary Angles

Two angles whose sum is 90°.
If one angle is $x$ , the other is $90^\circ - x$ .

2. Supplementary Angles

Two angles whose sum is 180°.
If one angle is $x$ , the other is $180^\circ - x$ .

3. Adjacent Angles

Two angles that share a common arm and vertex and do not overlap.

4. Linear Pair

Two adjacent angles that form a straight line.
Sum = 180°.

5. Vertically Opposite Angles

Formed when two lines intersect. The opposite (non-adjacent) angles are equal.

Angle Formed by Intersecting Lines

When two lines intersect:

Four angles are formed.
Each pair of vertically opposite angles are equal.
Each adjacent angle pair forms a linear pair and is supplementary.

If line AB intersects line CD at point O:

Then:

∠AOC = ∠BOD (vertically opposite)
∠AOD + ∠BOC = 180° (linear pair)

Parallel Lines and a Transversal

When a transversal cuts two parallel lines, the following angle relationships are formed:

Types of Angles Formed:

Type	Property
Corresponding Angles	Equal
Alternate Interior Angles	Equal
Alternate Exterior Angles	Equal
Consecutive Interior	Supplementary (sum = 180°)

Diagram:

    l1:  -------------------------
               \      
                \      Transversal
               /      
    l2:  -------------------------

Important Angle Facts

Sum of angles on a straight line = 180°
Sum of angles at a point = 360°
Vertically opposite angles are equal
If two lines are parallel, corresponding and alternate angles are equal

Key Formulas and Theorems

Angle sum on straight line:
$\angle A + \angle B = 180^\circ$
Angle sum at a point:
$\sum \text{All angles around a point} = 360^\circ$
Vertically opposite angles are equal
Parallel line rules with transversal:
- $\angle 1 = \angle 5$ (corresponding)
- $\angle 3 = \angle 6$ (alternate interior)
- $\angle 4 + \angle 5 = 180^\circ$ (co-interior)

Visual Understanding

Parallel Lines with Transversal

       A        B
    ---------//---------  ← Line l1
            /
           /    ← Transversal t
          /
    ---------//---------  ← Line l2
       C        D

∠A and ∠D (corresponding) are equal
∠B and ∠C (alternate interior) are equal
∠A + ∠C = 180° (co-interior)

Common Mistakes and Conceptual Tips

Mistake	Fix
Assuming all intersecting lines are perpendicular	Only perpendicular lines form right angles
Misidentifying alternate and corresponding angles	Learn their position with diagrams
Not using the 180° or 360° rules correctly	Always check whether it’s a line or a full turn
Thinking only adjacent angles add up to 180°	Remember: linear pairs are adjacent + form a straight line

Examples

Example 1:

Two angles form a linear pair. One is 3 times the other. Find both angles.

Solution:
Let angle be $x$ , then other is $3x$

x + 3x = 180 \Rightarrow 4x = 180 \Rightarrow x = 45^\circ

Angles: 45°, 135°

Example 2:

In the figure, two lines intersect. One angle is 70°. Find all angles formed.

Solution:

Vertically opposite angle = 70°
Adjacent angle = $180 - 70 = 110^\circ$
Opposite to 110° = 110°

Angles: 70°, 110°, 70°, 110°

Example 3:

A transversal cuts two parallel lines. One of the corresponding angles is 65°. Find all angles.

Solution:
All corresponding and alternate angles = 65°, others = 115° (since they are supplementary)