Question

are the triangles congruent? why or why not?

yes, \\(\angle a \cong \angle d\\) and \\(\overline{ab} \cong \overline{de}\\).
yes, they are congruent by either aas or asa.
no, \\(\angle c\\) is not congruent to any angle in \\(\delta def\\).
no, the congruent sides do not correspond.

Explanation:

Find the missing angle in the first triangle

Using the Triangle Geometry knowledge point, the sum of angles in \(\triangle ABC\) is \(180^\circ\).

$$ m\angle B = 180^\circ - (88^\circ + 33^\circ) = 59^\circ $$

Find the missing angle in the second triangle

Using the Triangle Geometry knowledge point, the sum of angles in \(\triangle DEF\) is \(180^\circ\).

$$ m\angle F = 180^\circ - (58^\circ + 33^\circ) = 89^\circ $$

Compare the corresponding angles

Using the Triangle Congruence Theorems knowledge point, we compare the interior angles of \(\triangle ABC\) and \(\triangle DEF\).

• The angles of \(\triangle ABC\) are \(33^\circ\), \(59^\circ\), and \(88^\circ\).
• The angles of \(\triangle DEF\) are \(33^\circ\), \(58^\circ\), and \(89^\circ\).

Since the angle measures do not match, \(\angle C\) (\(88^\circ\)) is not congruent to any angle in \(\triangle DEF\).

Answer:

• (A) Yes, \(\angle A \cong \angle D\) and \(\overline{AB} \cong \overline{DE}\).
• (B) Yes, they are congruent by either AAS or ASA.
• (C) No, \(\angle C\) is not congruent to any angle in \(\triangle DEF\). (Correct answer)
• (D) No, the congruent sides do not correspond.