Question

question 1 of 10
what else would need to be congruent to show that △abc≅△
def by the aas theorem?

given:
∠a ≅ ∠d
∠b ≅ ∠e

a. (overline{bc} cong overline{df})
b. (∠c cong ∠f)
c. (∠b cong ∠e)
d. (overline{ac} cong overline{df})

Explanation:

Step1: Recall AAS Theorem

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

Step2: Analyze Given Information

We are given that $\angle A\cong\angle D$ and $\angle B\cong\angle E$. We need to find the non - included side that should be congruent.

• For $\triangle ABC$ and $\triangle DEF$, the angles $\angle A$ and $\angle B$ in $\triangle ABC$ correspond to $\angle D$ and $\angle E$ in $\triangle DEF$. The non - included side for $\angle A$ and $\angle B$ in $\triangle ABC$ is $AC$, and the non - included side for $\angle D$ and $\angle E$ in $\triangle DEF$ is $DF$. So we need $\overline{AC}\cong\overline{DF}$.
• Let's analyze other options:
• Option A: $\overline{BC}\cong\overline{DF}$: $BC$ is not the non - included side for $\angle A$ and $\angle B$, so this is incorrect.
• Option B: $\angle C\cong\angle F$: This would be for AAA (which doesn't prove congruence) or if we already had two angles and were looking for the third, but we need a side for AAS, so this is incorrect.
• Option C: $\angle B\cong\angle E$ is already given, so this is not what we need.

Answer:

D. $\overline{AC}\cong\overline{DF}$