Vertical Angles
In short: Vertical angles sit across from each other where two straight lines intersect, and the two angles in any such facing pair always have exactly the same measure.
7th GradeGeometry7.G.B.5
What is Vertical Angles?
Two straight lines that cross carve out four angles, and the ones facing each other, meeting only at the crossing point, are called vertical angles. The word refers to the shared vertex, not to up and down.
Each facing pair matches in size. A \(63^{\circ}\) angle has a \(63^{\circ}\) partner straight across from it, and the remaining two measure \(117^{\circ}\) because neighbors along one line total \(180^{\circ}\). That equality survives for any pair of crossing lines.
How to spot it
- Two lines intersect in a picture and the question names angles that sit across from each other.
- Two angle expressions with a variable face each other at a crossing and you must set them equal.
- A scissors, a railroad crossing or an X-shaped brace is drawn as two crossing segments.
- You know one angle at a crossing and must fill in the other three.
The method
- Locate the crossing point and identify the angle directly opposite the one you know.
- Write that opposite angle with the same measure.
- For each neighbor, subtract the known angle from \(180^{\circ}\), since two neighbors lie on one line.
- With algebra, set the two opposite expressions equal and solve for the variable.
Vertical Angles: solved example
Problem
A window brace forms an X. Facing angles at its center are labeled \((4x + 7)^{\circ}\) and \((6x – 21)^{\circ}\). Solve for \(x\), then name the matching angles and the angle beside them.
Solution
Facing angles match, so \(4x + 7 = 6x – 21\). Then \(28 = 2x\) and \(x = 14\).
Each matching angle is \(4(14) + 7 = 63\). The one beside it shares a line with it and measures \(180 – 63 = 117\). Result: \(63^{\circ}\) twice and \(117^{\circ}\) twice.
Try it
Question 1
Two lines cross and one angle measures \(118^{\circ}\). Find the other three angles.
Show the answer
The opposite angle is \(118^{\circ}\). Each neighbor is \(180 – 118 = 62\), so the angles are \(118^{\circ}, 62^{\circ}, 118^{\circ}, 62^{\circ}\).
Question 2
Opposite angles at a crossing are \((2x + 30)^{\circ}\) and \((3x + 10)^{\circ}\). What is each angle?
Show the answer
\(2x + 30 = 3x + 10\) gives \(x = 20\), so each measures \(70^{\circ}\).
Vertical Angles vs. Adjacent Angles
Vertical angles face each other across the crossing and are always equal. Adjacent angles are neighbors that share a side, and on a line they total \(180^{\circ}\) instead of matching. In an X, each angle has one opposite partner and two neighbors, so check which relationship the question names.
Keep going
Lessons
Worksheets and quizzes
- Angle Relationships and Equations – Worksheet
- Angle Relationships and Equations – Quiz
- Angles in Triangles and Parallel Lines – Worksheet
- Angles in Triangles and Parallel Lines – Quiz
Related words
Whole dictionary: A–Z · 7th Grade · Geometry