Question

qs bisects ∠pqr.
11.
find ∠sqr.
12.
find ∠pqs
13.
find ∠pqr.

Explanation:

Step1: Recall the definition of an angle bisector

An angle bisector divides an angle into two equal parts.

Step2: Apply the definition to problem 11

Since \( \overrightarrow{QS} \) bisects \( \angle PQR \), then \( \angle SQR=\angle PQS \). Given \( \angle PQS = 40^{\circ} \), so \( \angle SQR=40^{\circ} \).

Step3: Apply the definition to problem 12

Since \( \overrightarrow{QS} \) bisects \( \angle PQR \), then \( \angle PQS=\angle SQR \). Given \( \angle SQR = 64^{\circ} \), so \( \angle PQS=64^{\circ} \).

Step4: Apply the definition to problem 13

Since \( \overrightarrow{QS} \) bisects \( \angle PQR \), then \( \angle PQR=\angle PQS+\angle SQR \). Given \( \angle PQS = 52^{\circ} \) and \( \angle SQR = 52^{\circ} \), so \( \angle PQR=52^{\circ}+52^{\circ}=104^{\circ} \).

Answer:

1. \( 40^{\circ} \)
2. \( 64^{\circ} \)
3. \( 104^{\circ} \)