Question

the asa (angle, side, angle) 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 asa?

options:

• ∠a ≅ ∠x; \\(\overline{ab} ≅ \overline{xy}\\); \\(\overline{ac} ≅ \overline{xz}\\)
• ∠a ≅ ∠x; \\(\overline{ab} ≅ \overline{xy}\\); ∠b ≅ ∠y
• ∠a ≅ ∠x; ∠b ≅ ∠y; ∠c ≅ ∠z
• ∠a ≅ ∠x; \\(\overline{bc} ≅ \overline{yz}\\); \\(\overline{ac} ≅ \overline{xz}\\)

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. The included side is the side between the two angles.

Step2: Analyze Option 1

For the first option: $\angle A\cong\angle X$; $\overline{AB}\cong\overline{XY}$; $\overline{AC}\cong\overline{XZ}$. This looks like SAS (Side - Angle - Side) because we have two sides and the included angle? Wait, no. $\angle A$ is between $\overline{AB}$ and $\overline{AC}$, but here we have $\overline{AB}\cong\overline{XY}$, $\overline{AC}\cong\overline{XZ}$ and $\angle A\cong\angle X$. Wait, actually, this is SAS (if we consider the sides around $\angle A$ and $\angle X$). But we need ASA, so this is not ASA.

Step3: Analyze Option 2

Second option: $\angle A\cong\angle X$; $\overline{AB}\cong\overline{XY}$; $\angle B\cong\angle Y$. Let's check the positions. In $\triangle ABC$, $\overline{AB}$ is between $\angle A$ and $\angle B$. In $\triangle XYZ$, $\overline{XY}$ is between $\angle X$ and $\angle Y$. So we have two angles ($\angle A$ and $\angle B$) and the included side ($\overline{AB}$) of $\triangle ABC$ congruent to two angles ($\angle X$ and $\angle Y$) and the included side ($\overline{XY}$) of $\triangle XYZ$. This fits the ASA criterion.

Step4: Analyze Option 3

Third option: $\angle A\cong\angle X$; $\angle B\cong\angle Y$; $\angle C\cong\angle Z$. This is AAA (Angle - Angle - Angle), which only shows similarity, not congruence. So this is not ASA.

Step5: Analyze Option 4

Fourth option: $\angle A\cong\angle X$; $\overline{BC}\cong\overline{YZ}$; $\overline{AC}\cong\overline{XZ}$. Let's see the sides. $\overline{AC}$ and $\overline{BC}$ with $\angle A$? $\angle A$ is not between $\overline{AC}$ and $\overline{BC}$ (the side between $\angle A$ and $\angle C$ is $\overline{AC}$? Wait, no. In $\triangle ABC$, $\angle A$ is between $\overline{AB}$ and $\overline{AC}$. Here we have $\overline{AC}\cong\overline{XZ}$, $\overline{BC}\cong\overline{YZ}$ and $\angle A\cong\angle X$. This is not ASA. It might be AAS (Angle - Angle - Side) or something else, but not ASA.

Answer:

$\boldsymbol{\angle A\cong\angle X; \overline{AB}\cong\overline{XY}; \angle B\cong\angle Y}$ (the second option, assuming the options are labeled as 1,2,3,4 with the second one being the one with $\angle A\cong\angle X; \overline{AB}\cong\overline{XY}; \angle B\cong\angle Y$)