Question

part 3 of 3
prove that quadrilateral abcd is
congruent to quadrilateral
efgh
complete the proof below
statement reason
∠a ≅ ∠e
∠abd ≅ ∠efh, ∠c ≅ ∠g,
bc ≅ fg, cd ≅ gh given
△bcd ≅ △fgh sas
bd ≅ fh cpctc
△abd ≅ △efh
abcd ≅ efgh
ade up of corresponding
es are congruent.
sas
aas
sss
asa

Explanation:

Step1: Analyze $\triangle ABD$ and $\triangle EFH$

We know $\angle A \cong \angle E$, $\angle ABD \cong \angle EFH$, and from $ \triangle BCD \cong \triangle FGH$ (CPCTC) we have $\overline{BD} \cong \overline{FH}$. So two angles and a non - included side are congruent.

Step2: Identify the congruence criterion

The AAS (Angle - Angle - Side) congruence criterion 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, the triangles are congruent. For $\triangle ABD$ and $\triangle EFH$, we have two angles ($\angle A \cong \angle E$, $\angle ABD \cong \angle EFH$) and a non - included side ($\overline{BD} \cong \overline{FH}$) congruent. So the congruence criterion for $\triangle ABD \cong \triangle EFH$ is AAS.
For the quadrilateral congruence, since $ABCD$ is made up of $\triangle ABD$ and $\triangle BCD$, and $EFGH$ is made up of $\triangle EFH$ and $\triangle FGH$, and we have $\triangle ABD \cong \triangle EFH$ and $\triangle BCD \cong \triangle FGH$, the quadrilaterals $ABCD$ and $EFGH$ are congruent because their corresponding triangles are congruent.

Answer:

For $\triangle ABD \cong \triangle EFH$: AAS
For $ABCD \cong EFGH$: A quadrilateral is congruent to another if it is made up of corresponding congruent triangles.