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.