Question

if $overrightarrow{np}perpoverrightarrow{lm}$, find $mangle jnm$.
a) $129^{circ}$
b) $131^{circ}$
c) $134^{circ}$
d) $138^{circ}$
e) $141^{circ}$

Explanation:

Step1: Use perpendicular - angle property

Since $\overrightarrow{NP}\perp\overrightarrow{LM}$, $\angle LNP = 90^{\circ}$.
We know that $\angle LNJ+\angle JNP=\angle LNP$. So, $(13x - 23)+(9x + 3)=90$.

Step2: Solve the equation for $x$

Combine like - terms: $13x+9x-23 + 3=90$, which simplifies to $22x-20 = 90$.
Add 20 to both sides: $22x=90 + 20=110$.
Divide both sides by 22: $x = 5$.

Step3: Find $\angle JNM$

$\angle JNM$ and $\angle LNJ$ are supplementary (a linear pair), so $\angle JNM=180^{\circ}-\angle LNJ$.
Substitute $x = 5$ into the expression for $\angle LNJ$: $\angle LNJ=13x-23=13\times5-23=65 - 23=42^{\circ}$.
Then $\angle JNM=180 - 42=138^{\circ}$.

Answer:

D. $138^{\circ}$