Write a proof.

Question

Given: AB · BE = CB · BD
Prove: ΔABC ΔDBE

Answer 1

To prove that triangles ABC and DBE are congruent, we will use the Side-Angle-Side (SAS) congruence criteria.

Since AB · BE = CB · BD, we can rewrite this as:

AB/BD = CB/BE

This shows that triangles ABD and CBE are similar by the Side-Side-Side (SSS) similarity criteria.

Next, we have angle B is a common angle for both triangles, so we have:

∠ABC ≅ ∠DBE

Therefore, by the SAS congruence criteria, triangles ABC and DBE are congruent.