# Sample Proof on Triangle Congruence Part 1

We have discussed triangle congruence and in this series, we are going to use the congruence theorems in order to prove that two triangles are congruent.

Given

$\overline{AB} \cong \overline{AE}$
$\overline{AC} \cong \overline{AD}$

Prove
$\overline{BD} \cong \overline{EC}$

Proof

We can see that there are two overlapping triangles: triangle $ADB$ and triangle $ACE$.

$\angle B \cong \angle E$ by the Isosceles Triangle Theorem. Triangle $ABE$ is an isosceles triangle since $\overline{AB} \cong \overline{AE}$ (given). Now, the Isosceles Triangle Theorem states that opposite angles of the congruent sides of an isosceles triangle are congruent so, $\angle B \cong \angle E$.

$\angle ADC \cong \angle ACD$ by the Isosceles Triangle Theorem (the explanation is the same same as above).

$\triangle ABD \cong \triangle AEC$ by AAS Congruence Theorem.

$\overline{BD} \cong \overline{EC}$ since corresponding parts of congruent triangles are congruent.

This is what we want to prove.