Question

In the given figure, if $PQ\parallel RS,\angle MXQ={135}^{\circ }$, and $\angle MYR={40}^{\circ }$, find the measure of $\angle XMY$.

Answer 1

Through point M, draw a line AB parallel to the line PQ.



$\angle QXM$ and $\angle XMB$ are interior angles on the same side of the transversal XM.

$\angle QXM+\angle XMB={180}^{\circ }$ (Sum of the interior angles on the same side of transversal XM is supplementary.)

${135}^{\circ }+\angle XMB={180}^{\circ }$

$\angle XMB={180}^{\circ }-{135}^{\circ }={45}^{\circ }$

$\angle YMB=\angle MYR={40}^{\circ }$ [Alternate interior angles are equal.]

$\angle XMY=\angle XMB+\angle BMY$

$\angle XMY={45}^{\circ }+{40}^{\circ }={85}^{\circ }$