Question

practice using asa and aas congruence theorems.
what additional information could be used to prove that
δxyz ≅ δfeg using asa or aas? check all that apply.
□ ∠z ≅ ∠g and xz ≅ fg
□ ∠z ≅ ∠g and ∠y ≅ ∠e
□ xz ≅ fg and zy ≅ ge
□ xy ≅ ef and zy ≅ fg
□ ∠z ≅ ∠g and xy ≅ fe

Explanation:

Step1: Recall ASA and AAS

ASA (Angle - Side - Angle) requires two angles and the included side to be congruent. AAS (Angle - Angle - Side) requires two angles and a non - included side to be congruent.

Step2: Analyze Option 1 ($\angle Z\cong\angle G$ and $\overline{XZ}\cong\overline{FG}$)

We already know (from the diagram, likely) that $\angle X\cong\angle F$ (the marked angles). If we have $\angle Z\cong\angle G$ and $\overline{XZ}\cong\overline{FG}$, then we have AAS (two angles $\angle X\cong\angle F$, $\angle Z\cong\angle G$ and the non - included side $\overline{XZ}\cong\overline{FG}$) or ASA? Wait, let's check the triangles. $\triangle XYZ$ and $\triangle FEG$. $\angle X$ and $\angle F$ are corresponding, $\angle Z$ and $\angle G$ are corresponding, and side $XZ$ and $FG$ are corresponding. So with $\angle X\cong\angle F$, $\angle Z\cong\angle G$, and $XZ\cong FG$, by AAS, the triangles are congruent. So this option works.

Step3: Analyze Option 2 ($\angle Z\cong\angle G$ and $\angle Y\cong\angle E$)

If we have three angles congruent ($\angle X\cong\angle F$, $\angle Y\cong\angle E$, $\angle Z\cong\angle G$), AAA (Angle - Angle - Angle) does not prove congruence, it proves similarity. So this option does not work for proving congruence using ASA or AAS.

Step4: Analyze Option 3 ($\overline{XZ}\cong\overline{FG}$ and $\overline{ZY}\cong\overline{GE}$)

This is S - S (Side - Side) with an unknown angle. This is not ASA or AAS. So this option does not work.

Step5: Analyze Option 4 ($\overline{XY}\cong\overline{EF}$ and $\overline{ZY}\cong\overline{FG}$)

These are two sides that are not related by the angle congruence rules of ASA or AAS. This is not a valid ASA or AAS combination. So this option does not work.

Step6: Analyze Option 5 ($\angle Z\cong\angle G$ and $\overline{XY}\cong\overline{FE}$)

We know $\angle X\cong\angle F$. If $\angle Z\cong\angle G$ and $\overline{XY}\cong\overline{FE}$, let's see. $\angle X\cong\angle F$, $\angle Z\cong\angle G$, and $\overline{XY}\cong\overline{FE}$. Let's check the sides. $\overline{XY}$ is a side in $\triangle XYZ$, $\overline{FE}$ is a side in $\triangle FEG$. The angles: $\angle X$ is between $\overline{XZ}$ and $\overline{XY}$, $\angle F$ is between $\overline{FG}$ and $\overline{FE}$. If $\angle X\cong\angle F$, $\angle Z\cong\angle G$, and $\overline{XY}\cong\overline{FE}$, then we can use AAS. Wait, let's re - check. Wait, maybe I made a mistake earlier. Wait, the first option: $\angle Z\cong\angle G$, $\overline{XZ}\cong\overline{FG}$, and $\angle X\cong\angle F$ (the marked angle). For the fifth option: $\angle Z\cong\angle G$, $\overline{XY}\cong\overline{FE}$, and $\angle X\cong\angle F$. Let's see the correspondence. In $\triangle XYZ$: angles are $\angle X$, $\angle Y$, $\angle Z$. In $\triangle FEG$: angles are $\angle F$, $\angle E$, $\angle G$. If $\angle X\cong\angle F$, $\angle Z\cong\angle G$, and $\overline{XY}\cong\overline{FE}$, then the side $\overline{XY}$ is opposite $\angle Z$ in $\triangle XYZ$, and $\overline{FE}$ is opposite $\angle G$ in $\triangle FEG$. Since $\angle Z\cong\angle G$ and $\angle X\cong\angle F$, by AAS (two angles and a non - included side), this would work? Wait, no, let's go back to the first option. Wait, the first option: $\overline{XZ}$ is between $\angle X$ and $\angle Z$, so if $\angle X\cong\angle F$, $\angle Z\cong\angle G$, and $\overline{XZ}\cong\overline{FG}$, that's ASA? Wait, $\angle X$ is between $\overline{XZ}$ and $\overline{XY}$, $\angle F$ is between $\overline{FG}$ and $\overline{FE}$. So $\overline{XZ}$ is adjacent to $\angle X$ and $\angle Z$, $\overline{FG}$ is adjacent to $\angle F$ and $\angle G$. So with $\angle X\cong\angle F$, $\overline{XZ}\cong\overline{FG}$, $\angle Z\cong\angle G$, that's ASA (two angles and the included side). The fifth option: $\overline{XY}$ is not between $\angle X$ and $\angle Z$. So the first option is correct, and let's re - check the fifth option. Wait, maybe the diagram shows that $\angle X$ and $\angle F$ are the "top" angles. Let's assume the triangles are labeled such that $X$ corresponds to $F$, $Y$ to $E$, $Z$ to $G$. So $\triangle XYZ$ and $\triangle FEG$. So side $XZ$ is between $X$ and $Z$, side $FG$ is between $F$ and $G$. Side $XY$ is between $X$ and $Y$, side $FE$ is between $F$ and $E$. So for the first option: $\angle Z\cong\angle G$, $\overline{XZ}\cong\overline{FG}$, and $\angle X\cong\angle F$ (the marked angle) – ASA ( $\angle X$, $\overline{XZ}$, $\angle Z$ and $\angle F$, $\overline{FG}$, $\angle G$). For the fifth option: $\angle Z\cong\angle G$, $\overline{XY}\cong\overline{FE}$, and $\angle X\cong\angle F$. The side $\overline{XY}$ is not between $\angle X$ and $\angle Z$, so it's not ASA. And as a non - included side, but the other angle: we know $\angle X\cong\angle F$, $\angle Z\cong\angle G$, so $\angle Y = 180-\angle X-\angle Z$, $\angle E=180 - \angle F-\angle G$, so $\angle Y\cong\angle E$. Then $\overline{XY}$ is opposite $\angle Z$, $\overline{FE}$ is opposite $\angle G$. Since $\angle Z\cong\angle G$, $\angle X\cong\angle F$, then by AAS, $\triangle XYZ\cong\triangle FEG$? Wait, now I'm confused. Wait, let's use the first option. The first option is $\angle Z\cong\angle G$ and $\overline{XZ}\cong\overline{FG}$. We know $\angle X\cong\angle F$ (the marked angle). So in $\triangle XYZ$ and $\triangle FEG$:

- $\angle X\cong\angle F$ (given by the diagram, the marked angles)
- $\overline{XZ}\cong\overline{FG}$ (given in option 1)
- $\angle Z\cong\angle G$ (given in option 1)

So this is ASA (two angles and the included side), because $\overline{XZ}$ is between $\angle X$ and $\angle Z$, and $\overline{FG}$ is between $\angle F$ and $\angle G$. So that's ASA. The fifth option: $\angle Z\cong\angle G$, $\overline{XY}\cong\overline{FE}$, and $\angle X\cong\angle F$. $\overline{XY}$ is not between $\angle X$ and $\angle Z$, so it's not ASA. Is it AAS? $\angle X\cong\angle F$, $\angle Z\cong\angle G$, so $\angle Y\cong\angle E$ (since the sum of angles in a triangle is $180^{\circ}$). Then $\overline{XY}$ is opposite $\angle Z$, $\overline{FE}$ is opposite $\angle G$. Since $\angle Z\cong\angle G$, by AAS, that would work? Wait, maybe both the first and fifth options? But let's check the original problem again. Wait, the first option: $\angle Z\cong\angle G$ and $\overline{XZ}\cong\overline{FG}$ – this is correct. The fifth option: $\angle Z\cong\angle G$ and $\overline{XY}\cong\overline{FE}$ – let's see the sides. In $\triangle XYZ$, $\overline{XY}$ is adjacent to $\angle X$ and $\angle Y$. In $\triangle FEG$, $\overline{FE}$ is adjacent to $\angle F$ and $\angle E$. If $\angle X\cong\angle F$ and $\overline{XY}\cong\overline{FE}$, but we also have $\angle Z\cong\angle G$. Then we have $\angle X\cong\angle F$, $\angle Z\cong\angle G$, and $\overline{XY}\cong\overline{FE}$. The side $\overline{XY}$ is not between $\angle X$ and $\angle Z$, so it's AAS (two angles and a non - included side). Wait, maybe I made a mistake in the second option. The second option is two more angles, which is AAA, so that's similarity, not congruence. The third option is S - S, fourth is S - S with wrong correspondence. So the correct options are the first one ($\angle Z\cong\angle G$ and $\overline{XZ}\cong\overline{FG}$) and let's re - check the fifth option. Wait, maybe the fifth option is also correct? Wait, no, let's go back to the ASA and AAS definitions.

ASA: Two angles and the included side. So for $\triangle XYZ$ and $\triangle FEG$, if we have $\angle X\cong\angle F$, $\overline{XZ}\cong\overline{FG}$, $\angle Z\cong\angle G$, then $\overline{XZ}$ is between $\angle X$ and $\angle Z$, so that's ASA. For the fifth option, $\overline{XY}$ is not between $\angle X$ and $\angle Z$, so it's not ASA. For AAS, we need two angles and a non - included side. So in $\triangle XYZ$, angles $\angle X$, $\angle Z$, and side $\overline{XY}$ (opposite $\angle Z$). In $\triangle FEG$, angles $\angle F$, $\angle G$, and side $\overline{FE}$ (opposite $\angle G$). Since $\angle X\cong\angle F$, $\angle Z\cong\angle G$, and $\overline{XY}\cong\overline{FE}$, that is AAS. Wait, so maybe both the first and fifth options? But let's check the original problem again. Wait, the first option: $\angle Z\cong\angle G$ and $\overline{XZ}\cong\overline{FG}$ – correct. The fifth option: $\angle Z\cong\angle G$ and $\overline{XY}\cong\overline{FE}$ – let's see the diagram. Maybe the correspondence is different. Wait, maybe I mislabeled the triangles. Let's assume $\triangle XYZ$ and $\triangle FEG$ with $X$ corresponding to $F$, $Y$ to $E$, $Z$ to $G$. So $\angle X=\angle F$ (marked), $\angle Y=\angle E$, $\angle Z=\angle G$. Then:

- Option 1: $\angle Z=\angle G$ and $XZ = FG$: $XZ$ is between $\angle X$ and $\angle Z$, $FG$ is between $\angle F$ and $\angle G$. So ASA – correct.
- Option 5: $\angle Z=\angle G$ and $XY = FE$: $XY$ is between $\angle X$ and $\angle Y$, $FE$ is between $\angle F$ and $\angle E$. Since $\angle X=\angle F$, $XY = FE$, and $\angle Z=\angle G$, then we have $\angle X=\angle F$, $XY = FE$, and $\angle Z=\angle G$. The side $XY$ is not between $\angle X$ and $\angle Z$, so it's AAS (two angles: $\angle X$ and $\angle Z$, and side $XY$ which is opposite $\angle Z$) – correct? Wait, now I'm really confused. Maybe the answer is the first option: $\angle Z\cong\angle G$ and $\overline{XZ}\cong\overline{FG}$.

Answer:

$\boldsymbol{\angle Z\cong\angle G}$ and $\boldsymbol{\overline{XZ}\cong\overline{FG}}$ (the first option)