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All courses Geometry · G-CO.9 21 of 57
Prove theorems about lines and angles: vertical angles, parallel-line angle relationships, and perpendicular bisectors.

Use vertical angles to find a missing angle measure or variable

Problem
Given \(\text{one}~\text{vertical}~\text{angle}\) is \(72^{\circ}\), use vertical angles to find the missing angle measure.
Unworked line or perpendicular-bisector configuration showing only supplied evidence. Open full size
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Hint

Identify the missing angle as the angle directly opposite the 72 degree angle. Opposite angles formed by intersecting lines are vertical angles.

Vertical angles are congruent, so they have the same measure. Do not subtract from 180 unless the angles are adjacent and form a linear pair.

Solution walkthrough

01

Identify the pair by position

\[\text{known}~\text{and}~\text{unknown}~\text{angles}~\text{lie}~\text{opposite}~\text{at}~\text{one}~\text{intersection}\]

Opposite nonadjacent regions formed by the same two intersecting lines are vertical angles.

02

Apply the vertical angles theorem

\[\text{vertical}~\text{angles}~\text{are}~\text{congruent}\]

The theorem gives equal measures, not a supplementary relationship.

03

Transfer the measure

\[\text{missing}~\text{angle}=72^{\circ}\]

The angle directly opposite the given 72-degree region has the same measure.

04

Distinguish the adjacent case

\[\text{adjacent}~\text{angles}=180-72=108^{\circ}\]

This check explains why 108 belongs to a linear-pair region, not the requested vertical angle.

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Another way

  1. Label the opposite regions 1 and 3 and use m angle 1=m angle 3.

!

Common mistake

Do not subtract from 180; that finds an adjacent angle rather than the opposite vertical angle.

Completed line or perpendicular-bisector diagram with theorem evidence and conclusion.
72°

Solution walkthrough video