In \triangle ABC, the measure of interior angle \angle A is 45^\circ, and the exterior ang… — ACT

In $\triangle ABC$ , the measure of interior angle $\angle A$ is $45^\circ$ , and the exterior angle at vertex $B$ measures $125^\circ$ . What is the measure, in degrees, of interior angle $\angle C$ ?

Answer: 80 degrees

Answer

The measure of interior angle

\angle C

80

degrees.

According to the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. In this case, the exterior angle at vertex

B

measures

125^\circ

, and one of its remote interior angles,

\angle A

, measures

45^\circ

. The other remote interior angle is

\angle C

. Setting up the equation:

125^\circ = 45^\circ + \text{m}\angle C

. Solving for the measure of

\angle C

gives

125^\circ - 45^\circ = 80^\circ

Step-by-Step Solution

Use the Exterior Angle Theorem to relate the given angles.

The exterior angle at vertex

B

(

125^\circ

) is equal to the sum of the remote interior angles,

\angle A

and

\angle C

. This gives the equation:

125^\circ = 45^\circ + \text{m}\angle C

The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

Solve the equation for the measure of

\angle C

\text{m}\angle C = 125^\circ - 45^\circ = 80^\circ

Subtract

45^\circ

from both sides of the equation to isolate the measure of

\angle C

Key Concept

Exterior Angle Theorem

Question

Answer

Step-by-Step Solution

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