Question

In the given proof, it is given that ab ∥ de. We need to prove that Δabc and Δedc are congruent triangles. The triangles a, b, c and e, d, c connect at point c. The sides ab and de are parallel. Which theorem can be used to prove the congruence of the triangles?

1) AA similarity theorem
2) ASA similarity theorem
3) AAS similarity theorem
4) SAS similarity theorem

Answer 1

To prove that triangles Δabc and Δedc are congruent, we can use the SAS (Side-Angle-Side) similarity theorem.

Step-by-step explanation:

To prove that triangles Δabc and Δedc are congruent, we can use the SAS (Side-Angle-Side) similarity theorem. This theorem states that if two triangles have two pairs of corresponding sides that are proportional and the included angles are congruent, then the triangles are congruent.

In this case, we are given that ab ∥ de, which means that segment ab is parallel to segment de. This implies that segments bc and cd are also parallel. Therefore, we have two pairs of corresponding sides (ab and de, and bc and cd) that are parallel and proportional.

Additionally, triangles Δabc and Δedc share the common side ac. Since the included angles bac and ecd are also congruent, we have fulfilled the angle requirement of the SAS similarity theorem.

Therefore, we can conclude that triangles Δabc and Δedc are congruent by the SAS similarity theorem.