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We are given that \( \overrightarrow{QS} \) is the angle bisector of \( \angle PQR \), and we are asked to find the measure of \( \angle SQR \).

### Step-by-step solution:

1. **Understand the bisector**:
   Since \( \overrightarrow{QS} \) is the bisector of \( \angle PQR \), it divides \( \angle PQR \) into two equal parts. That means:
   \[
   \angle PQS = \angle SQR
   \]
   
2. **Given information**:
   The measure of \( \angle PQR \) is given as \( 80^\circ \).

3. **Using the bisector property**:
   Since \( \overrightarrow{QS} \) bisects \( \angle PQR \), each of the two angles formed (\( \angle PQS \) and \( \angle SQR \)) is half of \( 80^\circ \). Therefore:
   \[
   \angle SQR = \frac{\angle PQR}{2} = \frac{80^\circ}{2} = 40^\circ
   \]

Thus, the measure of \( \angle SQR \) is \( 40^\circ \).

Would you like more details or have any questions about this solution?

Here are 5 related questions to deepen your understanding:
1. What is the definition of an angle bisector?
2. How would you calculate the measure of \( \angle SQR \) if \( \angle PQR \) was different, say \( 100^\circ \)?
3. What are some properties of angles formed by intersecting lines?
4. How can you verify that a ray bisects an angle?
5. If the measure of \( \angle SQR \) were given, how would you find \( \angle PQR \)?

**Tip**: Always remember that an angle bisector divides an angle into two equal parts, making calculations simpler.

QS is the angle bisector of ∠PQR. What is the measure of ∠SQR? Given ∠PQR = 80°.

Learn how to calculate the measure of ∠SQR when QS bisects ∠PQR, where ∠PQR = 80°. This problem covers key geometric concepts such as angle bisectors and is suitable for students in Grades 6-8.

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