Question

analyzing angle bisectors $overrightarrow{qs}$ is the angle bisector of $angle pqr$. find the two angle measures not given in the diagram.

Explanation:

Step1: Recall angle - bisector property

An angle - bisector divides an angle into two equal angles.

Step2: Find ∠PQS

Since $\overrightarrow{QS}$ is the angle - bisector of $\angle PQR$ and $\angle PQR = 75^{\circ}$, then $\angle PQS=\frac{1}{2}\angle PQR$. So, $\angle PQS=\frac{1}{2}\times75^{\circ}= 37.5^{\circ}$.

Step3: Find ∠SQR

$\angle SQR$ is equal to $\angle PQS$ because $\overrightarrow{QS}$ is the angle - bisector. So, $\angle SQR = 37.5^{\circ}$.

Answer:

$\angle PQS = 37.5^{\circ}$, $\angle SQR = 37.5^{\circ}$