Question

1. mark any congruent parts you can prove on the triangles. what piece of information would be required to prove the triangles congruent by aas? a) \\( \overline { e b } \cong \overline { d b } \\) b) \\( \angle a \cong \angle c \\) c) \\( \overline { a e } \cong \overline { c d } \\) d) \\( \angle e \cong \angle d \\)

Explanation:

Step1: Recall AAS (Angle - Angle - Side) congruence criterion

AAS states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the triangles are congruent.

Step2: Analyze the vertical angles

The vertical angles \(\angle ABE\) and \(\angle CBD\) are congruent (\(\angle ABE\cong\angle CBD\))

Step3: Check each option

• Option A: \(\overline{EB}\cong\overline{DB}\) is a side, but we need two angles and a non - included side. Just a side (without two angles) is not sufficient for AAS.
• Option B: If \(\angle A\cong\angle C\), along with the vertical angles \(\angle ABE\cong\angle CBD\), and if we have a non - included side. The non - included side for these two angles would be \(\overline{AE}\) and \(\overline{CD}\). But we need to check the side - angle - angle combination.
• Option C: \(\overline{AE}\cong\overline{CD}\). If we consider the two angles \(\angle ABE\cong\angle CBD\) (vertical angles) and assume \(\angle A\cong\angle C\) (from the AAS requirement of two angles), then \(\overline{AE}\) and \(\overline{CD}\) are non - included sides.
• Option D: \(\angle E\cong\angle D\) is only one angle (in addition to the vertical angles), we need two angles and a non - included side.

Answer:

C. \(\overline{AE}\cong\overline{CD}\)