Question

Given angle A, angle C , AE ≅ CD and angle BED ≅ angle BDE . Prove ΔAEB ≅ ΔCDB.

Answer 1

The triangle ΔAEB ≅ ΔCDB and it is prove to be true with the side angle sides SAS theorem.

Supplementary angles theorem:

Supplementary angles are angles the complement themselves in a triangle or parallel lines cut by a transversal and the sum of these supplementary angles are equal to 180 degrees.

Given that:

∠A = ∠C

AE ≅ CD

∠BED ≅ ∠BDE

From the triangle, angle AEB and angle BED are angles supplementary angles because the base of each angle at ∠E complement each other and are equal to 180 degrees.

Also, angle CDB and angle BDE are two acute angles and they are supplementary angle.

If that is in place and we know that the base angle of the isosceles triangle BED are equal;

Then, angle AEB and CDB are congruent triangles and their corresponding parts are the same (CPCTC) theorem.

Therefore, ΔAEB ≅ ΔCDB is proved according to the CPCTC and SAS theorem.

Answer 2

We are asked to prove that the triangles ΔAEB and ΔCDB are congruent.

We are given the following information.

∠A≅∠C

AE ≅ CD

∠BED≅∠BDE

2. Statement: ∠AEB and ∠BED are supplementary.

Notice that the angles AEB and BED form a linear pair meaning that their sum must be equal to 180°.

A linear pair is always supplementary (sum of angles is equal to 180°)

So, the correct reason is "If two angles form a linear pair, then they are supplementary"

3. Statement: ∠CDB and ∠BDE are supplementary.

Notice that the angles CDB and BDE form a linear pair meaning that their sum must be equal to 180°.

A linear pair is always supplementary (sum of angles is equal to 180°)

So, the correct reason is "If two angles form a linear pair, then they are supplementary"

4. Statement: ∠AEB ≅ ∠CDB

5. Statement: ΔAEB ≅ ΔCDB

Notice that we have two pairs of equal angles.

∠A≅∠C and ∠AEB ≅ ∠CDB

Also, we have one equal including side (includein