Question

mnpq is a trapezoid. find ( mangle pqm ).

Explanation:

Step1: Identify trapezoid parallel sides

$MN \parallel PQ$ (marked with congruent segment indicators)

Step2: Use alternate interior angles

$\angle NMP = \angle QPM = 32^\circ$ (alternate interior angles for parallel lines $MN \parallel PQ$ cut by transversal $MP$)

Step3: Analyze $\triangle MNP$

$MN = NP$ (marked congruent), so $\triangle MNP$ is isosceles. Thus $\angle NMP = \angle NPM = 32^\circ$.

Step4: Calculate $\angle MPQ$

$\angle MPQ = 32^\circ$ (from Step2)

Step5: Find $\angle NPQ$

$\angle NPQ = \angle NPM + \angle MPQ = 32^\circ + 32^\circ = 64^\circ$

Step6: Use isosceles trapezoid property

In trapezoid $MNPQ$, $MN \parallel PQ$ and $NP = MQ$ (marked congruent), so it is isosceles. $\angle PQM = \angle NPQ$

Answer:

$64^\circ$