Question

consider quadrilateral lmno. if quadrilateral lmno is a parallelogram, what must the measure of angle lmn be? m∠lmn =
l (25x)° m
o (22x + 9)° n

Explanation:

Step1: Recall property of parallelogram

In a parallelogram, adjacent angles are supplementary, so $\angle L + \angle N=180^{\circ}$.
$25x+(22x + 9)=180$.

Step2: Combine like - terms

$25x+22x+9 = 180$.
$47x+9=180$.

Step3: Isolate the variable term

Subtract 9 from both sides: $47x=180 - 9$.
$47x=171$.

Step4: Solve for x

$x=\frac{171}{47}$.

Step5: Find $\angle LMN$

Since $\angle LMN$ and $\angle L$ are adjacent, and $\angle L = 25x$, and $\angle LMN=180 - 25x$.
First, find $\angle L$: $\angle L=25\times\frac{171}{47}=\frac{4275}{47}$.
$\angle LMN=180-\frac{4275}{47}=\frac{8460 - 4275}{47}=\frac{4185}{47}\approx89.04^{\circ}$. But we made a wrong start above. Adjacent angles of a parallelogram are supplementary. Let's use the fact that opposite angles are equal. So $25x=22x + 9$.

Step6: Solve for x using opposite - angle property

Subtract $22x$ from both sides: $25x-22x=9$.
$3x=9$.
$x = 3$.

Step7: Find $\angle LMN$

Since $\angle LMN$ and $\angle L$ are adjacent, and $\angle L=25x$. When $x = 3$, $\angle L=25\times3 = 75^{\circ}$.
$\angle LMN=180 - 75=105^{\circ}$.

Answer:

$105$