Question

1. in the diagram shown, lines $overline{mn}$ and $overline{pq}$ intersect at r. if $angle nrs$ is a right - angle and the measure of $angle mrq$ is $142^{circ}$, then what is the measure of $angle prs$? show or explain how you arrive at your answer.

Explanation:

Step1: Identify vertical - angles

Vertical angles are equal. $\angle MRQ$ and $\angle NRP$ are vertical angles, so $\angle NRP=\angle MRQ = 142^{\circ}$.

Step2: Use the right - angle relationship

We know that $\angle NRS = 90^{\circ}$, and $\angle NRP=\angle NRS+\angle PRS$.
So, $\angle PRS=\angle NRP - \angle NRS$.

Step3: Calculate the measure of $\angle PRS$

Substitute the values: $\angle PRS=142^{\circ}-90^{\circ}=52^{\circ}$.

Answer:

$52^{\circ}$