Question

what else would need to be congruent to show that $\triangle abc \cong \triangle xyz$ by asa?
given:
$\angle z \cong \angle c$
$\overline{cb} \cong \overline{zy}$
a. $\angle y \cong \angle b$
b. $\overline{ac} \cong \overline{xz}$
c. $\overline{bc} \cong \overline{yz}$
d. $\angle z \cong \angle a$

Explanation:

Step1: Recall ASA Congruence

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 triangles are congruent.

Step2: Analyze Given Information

We are given that $\angle Z\cong\angle C$ and $\overline{CB}\cong\overline{ZY}$. In $\triangle ABC$ and $\triangle XYZ$, the side $\overline{CB}$ is between $\angle B$ and $\angle C$, and the side $\overline{ZY}$ is between $\angle Y$ and $\angle Z$. For ASA, we need the other pair of angles that include the given side to be congruent. So we need $\angle Y\cong\angle B$ because then we will have two angles ($\angle Y\cong\angle B$, $\angle Z\cong\angle C$) and the included side ($\overline{ZY}\cong\overline{CB}$) congruent, satisfying ASA.

Let's check the other options:

• Option B: $\overline{AC}\cong\overline{XZ}$ would be related to SSS or SAS, not ASA.
• Option C: $\overline{BC}\cong\overline{YZ}$ is the same as the given $\overline{CB}\cong\overline{ZY}$ (since $\overline{BC}$ and $\overline{CB}$ are the same segment, and $\overline{YZ}$ and $\overline{ZY}$ are the same segment), so this is already given.
• Option D: $\angle Z\cong\angle A$ does not help in forming the ASA criterion as $\angle Z$ is already related to $\angle C$.

Answer:

A. $\angle Y\cong\angle B$