Question

in the diagram below, ( mangle a=mangle d ), ( mangle b=mangle e ), and ( overline{ab}congoverline{de} ).
which of the following statements are true? select all that apply.
( \triangle abc ) and ( \triangle def ) are similar but not congruent.
( \triangle abc ) can be mapped onto ( \triangle def ) by a sequence of rigid transformations.
( overline{ac} ) is congruent to ( overline{df} ).
( angle c ) is congruent to ( angle f ).

Explanation:

Step1: Check triangle congruence

Given \(m\angle A = m\angle D\), \(m\angle B=m\angle E\), and \(AB = DE\). By the Angle - Side - Angle (ASA) congruence criterion (\(\triangle ABC\) and \(\triangle DEF\): \(\angle A=\angle D\), \(AB = DE\), \(\angle B=\angle E\)), \(\triangle ABC\cong\triangle DEF\).
Since \(\triangle ABC\cong\triangle DEF\), they are not just similar (similarity is a weaker condition, but here we have congruence). Rigid transformations (translations, rotations, reflections) can map one congruent triangle onto another.
If \(\triangle ABC\cong\triangle DEF\), then corresponding parts are congruent. For \(\triangle ABC\) and \(\triangle DEF\), \(\overline{AC}\) corresponds to \(\overline{DF}\) (by ASA congruence, the sides opposite the equal angles), so \(\overline{AC}\cong\overline{DF}\). Also, \(\angle C\) corresponds to \(\angle F\) (third angle of the triangle, since the sum of angles in a triangle is \(180^{\circ}\): \(\angle C=180^{\circ}-\angle A - \angle B\) and \(\angle F=180^{\circ}-\angle D-\angle E\), and \(\angle A=\angle D\), \(\angle B=\angle E\)), so \(\angle C\cong\angle F\).

Answer:

\(\triangle ABC\) can be mapped onto \(\triangle DEF\) by a sequence of rigid transformations, \(\overline{AC}\) is congruent to \(\overline{DF}\), \(\angle C\) is congruent to \(\angle F\)