Question

14
what else would need to be congruent to show that $\triangle abc \cong \triangle xyz$ by sas?
given
$\overline{ab} \cong \overline{xy}$
$\angle a \cong \angle x$
a. $\angle c \cong \angle z$
b. $\overline{bc} \cong \overline{yz}$
c. $\angle b \cong \angle y$
d. $\overline{ac} \cong \overline{xz}$

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 two triangles are congruent.

Step2: Analyze Given Information

We know that $\overline{AB}\cong\overline{XY}$ and $\angle A\cong\angle X$. For SAS, we need the side that is adjacent to $\angle A$ and $\overline{AB}$ in $\triangle ABC$, and the side adjacent to $\angle X$ and $\overline{XY}$ in $\triangle XYZ$ to be congruent. The side adjacent to $\angle A$ and $\overline{AB}$ in $\triangle ABC$ is $\overline{AC}$, and the side adjacent to $\angle X$ and $\overline{XY}$ in $\triangle XYZ$ is $\overline{XZ}$. So we need $\overline{AC}\cong\overline{XZ}$.

Step3: Evaluate Options

• Option A: $\angle C\cong\angle Z$ - This is related to AAS or ASA, not SAS.
• Option B: $\overline{BC}\cong\overline{YZ}$ - This does not form the included angle with the given congruent side and angle.
• Option C: $\angle B\cong\angle Y$ - This is related to ASA or AAS, not SAS.
• Option D: $\overline{AC}\cong\overline{XZ}$ - This provides the second side for the SAS criterion as it is the side included with $\angle A$ and $\overline{AB}$ (and $\angle X$ and $\overline{XY}$).

Answer:

D. $\overline{AC}\cong\overline{XZ}$