Question

1. in the figure below, ∠mnr measures 108°. ∠mnq measures 46°. ∠pnr measures 77°. what is the measure of ∠pnq?

a 15°
b 31°
c 48°

Explanation:

Step1: Observe angle - relationships

We know that \(\angle MNR=\angle MNQ+\angle QNR\) and \(\angle PNR = \angle PNQ+\angle QNR\). So, \(\angle PNQ=\angle PNR-\angle QNR\), and we first need to find \(\angle QNR\).
Since \(\angle MNR = 108^{\circ}\) and \(\angle MNQ=46^{\circ}\), then \(\angle QNR=\angle MNR - \angle MNQ\).
\(\angle QNR=108^{\circ}- 46^{\circ}=62^{\circ}\)

Step2: Calculate \(\angle PNQ\)

We know that \(\angle PNR = 77^{\circ}\) and we found \(\angle QNR = 62^{\circ}\).
Using the formula \(\angle PNQ=\angle PNR-\angle QNR\), we have \(\angle PNQ=77^{\circ}-62^{\circ}=15^{\circ}\)

Answer:

A. \(15^{\circ}\)