Definition
Vertical angles (also called vertically opposite angles in some regions) are the pairs of opposite angles created when two straight lines cross at a point. Each pair shares the same vertex but does not share a side; the two angles face one another across the intersection.
Basic properties
The most important property is congruence: each angle in a vertical pair has the same measure as its opposite angle. From two intersecting lines you get two pairs of vertical angles. Adjacent angles that form the other halves of the intersection are called linear pairs and are supplementary (their measures add to 180°).
- Congruence: vertical angles are equal in measure.
- Linear pairs: each angle and its adjacent neighbor sum to 180°.
- Notation: angles are often named by three letters, e.g. ∠AOB, where O is the vertex.
Why they are equal (sketch of proof)
A common proof uses supplementary pairs: if two lines intersect, one angle and its adjacent angle form a straight line and so sum to 180°. The adjacent angles to the opposite angle are the same supplementary angles, so the two opposite angles must be equal. For a visual explanation see a basic geometry glossary: definition and diagram or a stepwise proof: proof details.
Examples and use
Vertical angles appear frequently in algebraic geometry problems. For example, if one angle is given as 3x + 20 and its opposite as 2x + 50, set them equal (3x + 20 = 2x + 50) and solve for x. Vertical angles are a standard tool in geometric proofs and in solving angle-finding problems involving intersecting lines and transversals. For worked examples and classroom activities, see example problems and teaching resources.
Common confusions and remarks
Despite the name, "vertical" here does not refer to up or down; it derives from "vertex" (the shared corner). Vertical angles are distinct from complementary angles (which sum to 90°) and from consecutive interior/exterior angles that arise with parallel lines and a transversal.