Question

question 2 of 10
what else would need to be congruent to show that \\( \triangle a b c \cong \triangle d e f \\) by asa?
a. \\( \overline{b c} \cong \overline{e f} \\)
b. \\( \angle a \cong \angle d \\)
c. \\( \overline{a c} \cong \overline{d f} \\)
d. \\( \angle b \cong \angle e \\)

Explanation:

Step1: Recall the ASA (Angle - Side - Angle) congruence criterion

ASA states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Step2: Analyze the given information

We have \(AB = DE=10\) and \(\angle A=\angle D = 45^{\circ}\). The included side for \(\angle A\) and \(\angle B\) in \(\triangle ABC\) is \(AB\), and for \(\angle D\) and \(\angle E\) in \(\triangle DEF\) is \(DE\). To use ASA, we need another pair of angles.

Step3: Check each option

• Option A: \(\overline{BC}\cong\overline{EF}\) is a side - side - side (SSS) or side - angle - side (SAS) related (not ASA).
• Option B: \(\angle A\cong\angle D\) is already given.
• Option C: \(\overline{AC}\cong\overline{DF}\) is a side - side - side (SSS) or side - angle - side (SAS) related (not ASA).
• Option D: If \(\angle B\cong\angle E\), then in \(\triangle ABC\) and \(\triangle DEF\), we have \(\angle A\cong\angle D\), \(AB = DE\), and \(\angle B\cong\angle E\) (satisfying ASA: two angles and the included side).

Answer:

D. \(\angle B\cong\angle E\)