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

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

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

Proof 

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

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

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

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

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

This is what we want to prove.