Vertically Opposite Angles

Statement

When two straight lines cross, the opposite (or “vertical”) angles at the intersection are equal:

\[ \angle A = \angle C, \quad \angle B = \angle D \]

This is true for any pair of intersecting lines.

Why it’s true

• At an intersection, each pair of adjacent angles makes a straight line, so they add to 180°.
• If one angle is known, the one next to it is 180° minus that value.
• The angle opposite must then equal the first angle, since both complete the same straight line.

Recipe (how to use it)

1. Identify two straight lines crossing.
2. Find the given angle.
3. State that the angle directly opposite is equal.

Spotting it

Look for an “X” shape formed by two lines. The equal angles are opposite across the intersection.

Common pairings

• Often used with angles on a straight line = 180°.
• Can be combined with parallel line rules in more complex diagrams.

Mini examples

1. Given: One angle is 65°. Find: Opposite angle. Answer: 65°.
2. Given: One angle is 120°. Find: Opposite angle. Answer: 120°.

Pitfalls

• Confusing “opposite” with “adjacent”.
• Forgetting that only opposite pairs are equal, not all angles at the intersection.

Exam strategy

• Mark all four angles at the intersection.
• Use straight line (180°) and vertically opposite angle facts together.
• Check carefully which angle the question is asking for.

Summary

Vertically opposite angles are equal whenever two straight lines cross. This simple rule often unlocks more complex angle problems when used with other angle facts.