The diagram shows two intersecting lines with point $Q$ as the intersection. The angle $\angle PQR$ is labeled $(3x - 5)^\circ$, and the adjacent angle $\angle RQS$ is labeled $(x + 1)^\circ$.

These two angles are supplementary because they form a straight line, meaning their measures must add up to 180 degrees. Therefore, we can set up the following equation:

$(3x - 5)^\circ + (x + 1)^\circ = 180^\circ$

Let's solve for $x$:

$3x - 5 + x + 1 = 180$

$4x - 4 = 180$

$4x = 184$

$x = 46$

Now that we know $x = 46$, we can substitute this value back into the expression for $m \angle PQR$:

$m \angle PQR = 3x - 5 = 3(46) - 5 = 138 - 5 = 133^\circ$

Thus, the measure of $\angle PQR$ is $133^\circ$.

Would you like more details or have any questions?

Here are 5 related questions you might explore:

1. How do we know that $\angle PQR$ and $\angle RQS$ are supplementary?
2. What would happen if these angles were not supplementary?
3. Can we find the measure of $\angle RQS$ using the value of $x$?
4. How would the solution change if the angles formed a triangle instead of a straight line?
5. What is the general rule for supplementary angles?

Tip: Always check if the angles form a straight line or other relationships like vertical angles or angles in a triangle to set up your equation.