Key takeaway: Vertical angles are the pair of angles facing each other across the point where two lines intersect, and the two members of a pair are always equal in size.
Grade 77.G.B.5Geometry and Measurement
Definition of Vertical Angles
Vertical angles are the angles that sit directly across from each other when two straight lines cross, touching only at the crossing point. Two crossing lines carve the plane into four angles, which makes two such pairs. The word “vertical” has nothing to do with up and down; it refers to the shared vertex.
The key property is that vertical angles are congruent. Picture a pair of scissors: as the blades open and close, the angle at the top and the angle at the bottom grow or shrink together. Angles that sit beside each other on one line are a different story: they trade off, so a wider one leaves less room for the other.
Formula and notation
Number the four angles \(\angle 1\) through \(\angle 4\) going around the crossing. Opposite members are equal:
\[m\angle 1 = m\angle 3 \qquad \text{and} \qquad m\angle 2 = m\angle 4\]
Two neighbors fill out half a turn together, so their sum is \(180^{\circ}\).
Key facts
- Two crossing lines always create exactly two pairs of equal opposite angles.
- Each of the four angles forms a straight-line pair with both neighbors.
- One right angle in the crossing forces all four to be right angles.
- The equality holds however the lines are tilted or labeled.
Vertical Angles example
Two lines cross. One angle measures \((3x + 10)^{\circ}\) and the angle opposite it measures \((5x – 24)^{\circ}\). Find \(x\), then find the measure of an angle next to them.
-
Opposite angles are equal: \(3x + 10 = 5x – 24\).
-
Solve: \(34 = 2x\), so \(x = 17\).
-
Each of the pair measures \(3(17) + 10 = 61\) degrees.
-
A neighbor is supplementary: \(180 – 61 = 119\) degrees.
\(x = 17\); the opposite pair measures \(61^{\circ}\) and the neighbors measure \(119^{\circ}\).
Common mistake
Watch out
Looking at the crossing above, Dana says the \(61^{\circ}\) angle and the \(119^{\circ}\) angle next to it are vertical angles because they share a vertex.
Why it is wrong
Sharing a vertex is not enough. Vertical angles must be directly across from each other, with no side in common. The \(61^{\circ}\) and \(119^{\circ}\) angles lean on a common ray and add to \(180^{\circ}\), so they are a supplementary pair. The match for \(61^{\circ}\) is the other \(61^{\circ}\) angle.
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Geometry and Measurement