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All courses Geometry · G-CO.10 19 of 57
Prove triangle theorems including angle sum, isosceles base angles, midsegment theorem, and medians concurrence.

Find a missing interior angle in a triangle using the 180° angle sum

Problem
In a triangle, \(\text{two}~\text{angles}\) measure \(45^{\circ}\) and \(65^{\circ}\). Find the third angle.
Unworked triangle with two interior angles labeled 45° and 65° and the third angle marked unknown. Open full size
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Hint

Use the triangle angle-sum rule: the three interior angles add to \(180\) degrees.

Missing angle = \(180^\circ\) minus the sum of the two known angles.

Solution walkthrough

01

Apply the triangle angle-sum theorem

\[45^{\circ}+65^{\circ}+x=180^{\circ}\]

Every triangle's interior angles sum to 180 degrees. The inspected diagram places 45 degrees at A, 65 degrees at B, and the unknown x at C.

02

Combine the known angles

\[45^{\circ}+65^{\circ}=110^{\circ}\]

The two supplied interior angles account for 110 degrees of the total.

03

Solve for the third angle

\[x=180^{\circ}-110^{\circ}=70^{\circ}\]

Subtracting the known total from 180 leaves the measure of angle C.

04

Check all three angles

\[45^{\circ}+65^{\circ}+70^{\circ}=180^{\circ}\]

The three measures reconstruct the required triangle sum, so the third angle is 70 degrees.

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Another way

  1. Compute x=180-(45+65) in one expression.

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Common mistake

Do not report 110 degrees. That is the sum of the known angles; the missing angle is the remainder from 180 degrees.

Answer triangle diagram: two interior angles are 45° and 65°; the third is 180°−45°−65°=70°.
70°

Solution walkthrough video