Question

question 7 of 10
what else would need to be congruent to show that $\triangle abc \cong \triangle xyz$ by sas?

image of two triangles, $\triangle abc$ with vertices a, b, c and $\triangle xyz$ with vertices x, y, z. given: $\overline{ab} \cong \overline{xy}$, $\overline{bc} \cong \overline{yz}$

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 SAS Congruence Criterion

The 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 two triangles are congruent.

Step2: Analyze the Given Information

We are given that \(\overline{AB}\cong\overline{XY}\) and \(\overline{BC}\cong\overline{YZ}\). For the SAS criterion, the angle between the two given sides (\(\overline{AB}\) and \(\overline{BC}\) in \(\triangle ABC\), and \(\overline{XY}\) and \(\overline{YZ}\) in \(\triangle XYZ\)) should be congruent. The angle between \(\overline{AB}\) and \(\overline{BC}\) is \(\angle B\), and the angle between \(\overline{XY}\) and \(\overline{YZ}\) is \(\angle Y\). So we need \(\angle B\cong\angle Y\).

Let's analyze the other options:

• Option A: \(\overline{BC}\cong\overline{YZ}\) is already given, so this is not what we need to show.
• Option B: \(\overline{AC}\cong\overline{XZ}\) would be for SSS (Side - Side - Side) congruence, not SAS.
• Option D: \(\angle C\cong\angle Z\) is not the included angle between the given sides.

Answer:

C. \(\angle B\cong\angle Y\)