Question

1. find the m∠d. what theorem did you use? theorem: third angles theorem

Explanation:

Step1: Recall Third Angles Theorem

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

Step2: Identify corresponding angles

In \(\triangle ABC\) and \(\triangle DEF\), \(\angle B = \angle E=63^{\circ}\) and \(\angle C=\angle F = 32^{\circ}\).

Step3: Calculate \(\angle A\)

In \(\triangle ABC\), using the angle - sum property of a triangle (\(\angle A+\angle B+\angle C = 180^{\circ}\)), so \(\angle A=180^{\circ}-(63^{\circ}+32^{\circ})=180^{\circ}-95^{\circ} = 85^{\circ}\).

Step4: Find \(\angle D\)

By the Third Angles Theorem, since \(\angle A\) and \(\angle D\) are the third - pair of angles, \(\angle D=\angle A = 85^{\circ}\).

Answer:

\(m\angle D = 85^{\circ}\)