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.

triangle congruence

\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.

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