Question

complete the proof that $\triangle efh \cong \triangle eig$.

statement	reason
2 $\angle feg \cong \angle hei$	given
3 $\angle ehf \cong \angle egi$	given
4 $m\angle gei = m\angle geh + m\angle hei$	additive property of angle
5 $m\angle feh = m\angle feg + m\angle geh$	additive property of angle
6 $m\angle gei = m\angle geh + m\angle feg$	substitution
7 $m\angle feh = m\angle gei$	transitive property of equa
8 $\triangle efh \cong \triangle eig$

Explanation:

Step1: Identify known congruences

We know \( \overline{EH} \cong \overline{EG} \) (side), \( \angle EHF \cong \angle EGI \) (angle), and from steps 6 - 7, \( \angle FEH \cong \angle GEI \) (angle).

Step2: Apply AAS congruence

In triangles \( \triangle EFH \) and \( \triangle EIG \):

• \( \angle FEH \cong \angle GEI \) (proven)
• \( \angle EHF \cong \angle EGI \) (given)
• \( \overline{EH} \cong \overline{EG} \) (given)

This satisfies the Angle - Angle - Side (AAS) Congruence Theorem, which states that if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, then the two triangles are congruent.

Answer:

Angle - Angle - Side (AAS) Congruence Theorem