Which two triangles are congruent by the AAS Theorem? Complete the congruence statement.

Steps to Solve

1. Understand the AAS Theorem

The Angle-Angle-Side (AAS) Theorem 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. A non-included side is a side that is not between the two angles.

2. Compare $\triangle FHG$ and $\triangle EGD$

We need to compare the angles and sides of the two triangles to see if they are congruent by AAS. $\angle H \cong \angle E$ (given), also line $FG \cong CD$ And $\angle G \cong \angle D$ (given). Thus, $\triangle FHG \cong \triangle EGD$ by the AAS Theorem.

3. Compare $\triangle FHG$ and $\triangle QRP$

We need to compare the angles and sides of the two triangles to see if they are congruent by AAS. $\angle H \cong \angle P$ (given), also line $FG \cong PR$ And $\angle G$ is not congruent to $\angle R$. Thus, $\triangle FHG$ and $\triangle QRP$ are not congruent by the AAS Theorem.

4. Compare $\triangle EGD$ and $\triangle QRP$

We need to compare the angles and sides of the two triangles to see if they are congruent by AAS. $\angle E \cong \angle P$ (given), also line $CD \cong PR$ And $\angle D$ is not congruent to $\angle R$. Thus, $\triangle EGD$ and $\triangle QRP$ are not congruent by the AAS Theorem.

5. Write the congruence statement

Since $\triangle FHG$ and $\triangle EGD$ are congruent, we write the congruence statement. We must make sure that the vertices are listed in the correct order, according to the corresponding angles. Since $\angle H \cong \angle E$, $\angle G \cong \angle D$, and $\angle F \cong \angle C$, $\triangle FHG \cong \triangle EGD$.