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. ( angle c cong angle f )

c. ( angle b cong angle 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 \(\angle B\cong\angle E\). For AAS, we need a non - included side. Let's analyze the sides:

• In \(\triangle ABC\) and \(\triangle DEF\), the angles \(\angle A\) and \(\angle B\) are in \(\triangle ABC\), and \(\angle D\) and \(\angle E\) are in \(\triangle DEF\). The non - included side between \(\angle A\) and \(\angle B\) is \(AC\), and the non - included side between \(\angle D\) and \(\angle E\) is \(DF\). So we need \(AC\cong DF\) (i.e., \(\overline{AC}\cong\overline{DF}\)).
• Let's analyze option A: \(\overline{BC}\cong\overline{DF}\): \(BC\) is included between \(\angle B\) and \(\angle C\), so this is not the non - included side for AAS with the given angles.
• Option B: \(\angle C\cong\angle F\): This would be AAA (if we only had angles) or part of ASA, not AAS.
• Option C: \(\angle B\cong\angle E\) is already given, so this is not new information for AAS.
• Option D: \(\overline{AC}\cong\overline{DF}\): As we reasoned, this is the non - included side for the given two angles, so it satisfies AAS.

Answer:

D. \(\overline{AC}\cong\overline{DF}\)