The given problem provides a geometric figure where $QPT$ and $RPS$ are straight lines, and $QR \parallel ST$. The angles involved are marked as $\angle QRP = 68^\circ$, $\angle PTS = 42^\circ$, and you are asked to find the measure of $\angle QPS$.

Step-by-Step Solution:

1. Since $QR \parallel ST$, the corresponding angles property holds. This means that the angle formed by $\overrightarrow{RQ}$ and $\overrightarrow{PS}$ will equal the angle formed by $\overrightarrow{ST}$ and $\overrightarrow{RP}$.

2. $\angle PTS = 42^\circ$ and $\angle QPS$ are corresponding angles, so $\angle QPS = 42^\circ$.

Thus, the measure of $\angle QPS$ is $42^\circ$.

Would you like more details or have any other questions?

Here are 5 related questions to expand on this topic:

1. How can we use alternate interior angles in parallel lines to solve similar problems?
2. What are other geometric properties involving corresponding angles?
3. Can you find the sum of all angles in quadrilateral $QRST$?
4. How do we determine whether two lines are parallel using angle relationships?
5. Can the transversal properties be applied if the lines are not parallel?

Tip: Always remember that when two lines are parallel, corresponding angles are congruent, which can greatly simplify angle calculations.