Question

consider quadrilateral lmno. if quadrilateral lmno is a parallelogram, what must the measure of angle lmn be? m∠lmn =

Explanation:

Step1: Recall parallelogram property

In a parallelogram, adjacent angles are supplementary, so $\angle LMN+\angle MNO = 180^{\circ}$. Also, $\angle LMN$ and $\angle MNO$ are adjacent angles, and we know that $25x+22x + 9=180$.

Step2: Combine like - terms

Combine the $x$ terms: $(25x+22x)+9 = 180$, which simplifies to $47x+9 = 180$.

Step3: Solve for $x$

Subtract 9 from both sides: $47x=180 - 9=171$. Then $x=\frac{171}{47}$.

Step4: Find $\angle LMN$

Since $\angle LMN = 25x$, substitute $x=\frac{171}{47}$ into it. $\angle LMN=25\times\frac{171}{47}=\frac{4275}{47}\approx91^{\circ}$. But we made a wrong start. In a parallelogram, opposite angles are equal. So $\angle L=\angle N$. Then $25x=22x + 9$.

Step5: Solve the new equation for $x$

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

Step6: Calculate $\angle LMN$

Adjacent angles of a parallelogram are supplementary. $\angle LMN$ and $\angle L$ are adjacent. $\angle L=25x$, with $x = 3$, $\angle L = 75^{\circ}$. Then $\angle LMN=180 - 75=105^{\circ}$.

Answer:

$105$