Question

the sas (side-angle-side) relationship is a way to show that triangles are congruent. sets of triangle parts are listed. which set gives parts that allow triangle abc to be proven congruent to triangle xyz by sas?

(image of triangles abc and xyz)

• $\angle a \cong \angle x$; $\overline{bc} \cong \overline{yz}$; $\overline{ac} \cong \overline{xz}$
• $\angle a \cong \angle x$; $\angle b \cong \angle y$; $\angle c \cong \angle z$
• $\angle a \cong \angle x$; $\overline{ab} \cong \overline{xy}$; $\overline{ac} \cong \overline{xz}$
• $\angle a \cong \angle x$; $\angle b \cong \angle y$; $\overline{ab} \cong \overline{xy}$

Explanation:

Step1: Recall SAS Congruence

SAS (Side - Angle - Side) congruence 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 triangles are congruent. The included angle is the angle between the two sides.

Step2: Analyze Option 1

For the first option: $\angle A\cong\angle X$; $\overline{BC}\cong\overline{YZ}$; $\overline{AC}\cong\overline{XZ}$. The angle $\angle A$ is not the included angle between $\overline{BC}$ and $\overline{AC}$ (the included angle for $\overline{BC}$ and $\overline{AC}$ in $\triangle ABC$ is $\angle C$), so this does not satisfy SAS.

Step3: Analyze Option 2

The second option: $\angle A\cong\angle X$; $\angle B\cong\angle Y$; $\angle C\cong\angle Z$. This is AAA (Angle - Angle - Angle), which shows similarity, not congruence (unless there is a side - length relation, but here only angles are given), so it does not satisfy SAS.

Step4: Analyze Option 3

The third option: $\angle A\cong\angle X$; $\overline{AB}\cong\overline{XY}$; $\overline{AC}\cong\overline{XZ}$. In $\triangle ABC$, $\angle A$ is the included angle between $\overline{AB}$ and $\overline{AC}$. In $\triangle XYZ$, $\angle X$ is the included angle between $\overline{XY}$ and $\overline{XZ}$. So we have two sides ($\overline{AB}\cong\overline{XY}$, $\overline{AC}\cong\overline{XZ}$) and the included angle ($\angle A\cong\angle X$) congruent, which satisfies SAS.

Step5: Analyze Option 4

The fourth option: $\angle A\cong\angle X$; $\angle B\cong\angle Y$; $\overline{AB}\cong\overline{XY}$. This is ASA (Angle - Side - Angle) congruence, not SAS.

Answer:

$\boldsymbol{\angle A\cong\angle X; \overline{AB}\cong\overline{XY}; \overline{AC}\cong\overline{XZ}}$ (the third option among the given choices)