Vertically Opposite Angles — Definition, Formula & Examples
Vertically opposite angles are the pairs of angles that sit across from each other when two straight lines cross. They are always equal in measure.
When two lines intersect at a point, they form two pairs of non-adjacent angles. Each pair consists of vertically opposite angles (also called vertical angles), and the angles in each pair are congruent.
How It Works
When two straight lines cross, they create four angles at the intersection point. The angles that are directly across from each other — not side by side — are vertically opposite. You can identify them because they share only the vertex, not a common side. Since the two angles in each pair combine with the same adjacent angle to form a straight line (180°), they must be equal to each other.
Worked Example
Problem: Two lines intersect, and one of the four angles formed measures 65°. Find the other three angles.
Find the vertically opposite angle: The angle directly across from the 65° angle is vertically opposite, so it equals 65°.
Find an adjacent angle: Any angle adjacent to the 65° angle forms a straight line with it, so they add up to 180°.
Find the remaining angle: The last angle is vertically opposite to the 115° angle, so it also equals 115°.
Answer: The four angles are 65°, 115°, 65°, and 115°.
Why It Matters
Vertically opposite angles appear constantly in proofs involving parallel lines cut by a transversal. Recognizing them quickly helps you solve angle problems in geometry courses and is essential for understanding triangle and polygon angle relationships.
Common Mistakes
Mistake: Confusing vertically opposite angles with adjacent angles at the same intersection.
Correction: Adjacent angles share a common side and add to 180°. Vertically opposite angles are across from each other and are equal. Check whether the two angles share a side — if they do, they are adjacent, not vertically opposite.