Question

1. if np bisects ∠mnq, m∠mnq=(8x + 8)°, m∠pnq = 84°, and m∠rnm=(3y - 9)°, find the values of x and y.

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{NP}$ bisects $\angle MNQ$, we have $m\angle MNP=m\angle PNQ$. So, $8x + 8=84$.

Step2: Solve for $x$

Subtract 8 from both sides of the equation $8x + 8=84$: $8x=84 - 8=76$. Then divide both sides by 8, $x=\frac{76}{8}=\frac{19}{2}=9.5$.

Step3: Use angle - relationship

We assume $\angle RNM$ and $\angle MNQ$ are supplementary (a common angle - relationship in such set - ups, if they form a linear pair). So, $m\angle RNM + m\angle MNQ = 180^{\circ}$. Since $m\angle MNQ = 2\times84^{\circ}=168^{\circ}$, then $3y-9+168 = 180$.

Step4: Solve for $y$

First simplify the left - hand side of the equation $3y-9 + 168=180$ to get $3y+159 = 180$. Subtract 159 from both sides: $3y=180 - 159 = 21$. Divide both sides by 3, $y = 7$.

Answer:

$x = 9.5,y = 7$