Vertically Opposite Angles | Geometry

Learning Objectives

identify vertically opposite angles;
use their equality to find missing angles;
combine vertically opposite and straight-line rules;
solve algebraic angle problems at intersections;
give reasons for angle answers.

Introduction

When two straight lines cross, they form four angles.

The crossing creates a balanced pattern: angles directly across the intersection match, while neighbouring angles complete a straight line.

Vertically opposite angles are equal.

Since the marked angles are vertically opposite, x = 72°.

Key Notes

One angle at an intersection is 64°.

The opposite angle is also 64°.

Each adjacent angle is 180°−64°=116°.

So the four angles are 64°, 116°, 64°, 116°.

Algebraic Example

Vertically opposite angles are (5x−7)° and (3x+29)°.

5x−7=3x+29

2x=36, so x=18.

Each angle is 83°.

How to Recognise the Pair

Vertically opposite angles are directly across the intersection. Adjacent angles sit next to each other and total 180°.

Try It: Vertically Opposite Angles

Choose one angle: 65°

Opposite angles are equal: 65° = 65°. Adjacent angles are supplementary: 65° + 115° = 180°.

🧠 Let’s Think Together

If one angle at an intersection is 64°, which angle can you find without calculating? The angle directly opposite is also 64°. Each neighbouring angle then equals 180° − 64° = 116°.

Worked Examples

Example 1: Find all four angles

One angle is 72°. Its vertically opposite angle is 72°. Each adjacent angle is 180° − 72° = 108°. The four angles are 72°, 108°, 72° and 108°.

Example 2: Use algebra

Vertically opposite angles are (5x − 7)° and (3x + 29)°. Set them equal: 5x − 7 = 3x + 29. Therefore 2x = 36 and x = 18.

Practice Questions

Foundation

One angle at an intersection is 48°. Find the other three angles.
Vertically opposite angles are x and 112°. Find x.

Developing

Vertically opposite angles are (4x+5)° and (6x−31)°. Find x.
One angle is (3x+12)° and its adjacent angle is (5x−8)°. Find x.

Challenge

The acute and obtuse angles at an intersection differ by 54°. Find all four angles.

✅ Check Your Work

Show practice answers

48°, 132°, 48°, 132°.
112°.
4x+5=6x−31, so x=18.
8x+4=180, so x=22. Angles: 78° and 102°.
Let acute=a and obtuse=a+54. Then 2a+54=180, so a=63°. Angles: 63°,117°,63°,117°.

Common Mistakes

Calling adjacent angles vertically opposite.
Assuming all four angles at an intersection are equal.
Using 360° when only a neighbouring straight-line pair is needed.

Exam Focus

State “vertically opposite angles are equal” when you use the rule. Mark the pair you are comparing, and use the straight-line rule separately for adjacent angles.

Summary

Vertically opposite angles are equal.
Adjacent angles on a straight line total 180°.
At an intersection, only two different angle sizes usually occur.