Question

question 3 of 10
what else would need to be congruent to show that $\triangle abc \cong \triangle xyz$ by asa?

given:
$\overline{ac} \cong \overline{xz}$
$\angle a \cong \angle x$

a. $\overline{bc} \cong \overline{yz}$
b. $\overline{ac} \cong \overline{xz}$
c. $\angle b \cong \angle y$
d. $\angle c \cong \angle z$

Explanation:

Step1: Recall ASA Criterion

ASA (Angle - Side - Angle) congruence criterion 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 Given Information

We are given that $\overline{AC}\cong\overline{XZ}$ and $\angle A\cong\angle X$. In $\triangle ABC$ and $\triangle XYZ$, the side $AC$ is between $\angle A$ and $\angle C$, and the side $XZ$ is between $\angle X$ and $\angle Z$. For ASA, we need the other pair of angles (the angles adjacent to the given side and including the given angle) to be congruent. So we need $\angle C\cong\angle Z$ because then we will have $\angle A\cong\angle X$, $\overline{AC}\cong\overline{XZ}$, and $\angle C\cong\angle Z$ which satisfies ASA.

Let's analyze the other options:

• Option A: $\overline{BC}\cong\overline{YZ}$ would be related to SSA or other criteria, not ASA.
• Option B: $\overline{AC}\cong\overline{XZ}$ is already given, so it's not what we need to show for ASA.
• Option C: $\angle B\cong\angle Y$ would be for AAS (Angle - Angle - Side) if combined with other given, not ASA.

Answer:

D. $\angle C\cong\angle Z$