In \triangle PQR, the measure of \angle P is 50^\circ, and the measure of the exterior ang… — ACT

In $\triangle PQR$ , the measure of $\angle P$ is $50^\circ$ , and the measure of the exterior angle at vertex $Q$ is $110^\circ$ . If the bisector of $\angle PRQ$ intersects side $PQ$ at point $S$ , what is the measure, in degrees, of $\angle PRS$ ?

Answer: 30 degrees

Answer

The measure of

\angle PRS

30^\circ

By using the linear pair relationship, the interior angle

\angle PQR

is found to be

70^\circ

. Applying the triangle angle sum theorem, the third interior angle

\angle PRQ

180^\circ - (50^\circ + 70^\circ) = 60^\circ

. The angle bisector

RS

divides this angle into two equal parts, resulting in a measure of

30^\circ

for

\angle PRS

Step-by-Step Solution

Find the interior angle at vertex

Q

\angle PQR = 70^\circ

The interior and exterior angles at a vertex are supplementary, summing to

180^\circ

Find the measure of interior angle

\angle PRQ

\angle PRQ = 60^\circ

The sum of the interior angles in any triangle is

180^\circ

Calculate the measure of the bisected angle

\angle PRS

\angle PRS = 30^\circ

An angle bisector divides the angle into two equal measures.

Key Concept

Triangle Angle Sum Theorem and Exterior Angle Relationships

Estimated Time:1m 30s

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