Question

what else would need to be congruent to show that $\triangle abc \cong \triangle def$ by sas?
a. $\overline{bc} \cong \overline{ef}$
b. $\overline{ac} \cong \overline{df}$
c. $\angle a \cong \angle d$
d. $\angle c \cong \angle f$

Explanation:

Step1: Recall SAS Congruence

SAS (Side - Angle - Side) congruence criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Step2: Analyze Given Information

We know that \(AB\cong DE\) (one pair of sides) and \(\angle A\cong\angle D\) (the included angle). For SAS, we need the other pair of sides that form the included angle with the given side. In \(\triangle ABC\), the sides forming \(\angle A\) are \(AB\) and \(AC\). In \(\triangle DEF\), the sides forming \(\angle D\) are \(DE\) and \(DF\). So we need \(AC\cong DF\) to satisfy SAS.

Let's check the options:

• Option A: \(BC\cong EF\) - This is not the side forming the included angle with the given side, so it doesn't satisfy SAS.
• Option B: \(AC\cong DF\) - This is the required side to satisfy SAS as it is the other side forming the included angle with the given congruent side and angle.
• Option C: \(\angle A\cong\angle D\) - This is already given, not what we need for SAS (we need a side).
• Option D: \(\angle C\cong\angle F\) - This is an angle, not a side, so it doesn't satisfy SAS.

Answer:

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