Question

which two triangles are congruent by the aas theorem? complete the congruence statement.
(images of three triangles: \triangle fhg, \triangle yxz, \triangle qpr with angle and side markings)
\triangle \boxed{} \cong \triangle \boxed{}

Explanation:

Step1: Recall AAS Theorem

AAS (Angle - Angle - Side) Theorem states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.

Step2: Analyze the triangles

- For $\triangle FHG$: We have one marked angle at $G$, one right - angled (or congruent marked angle) at $H$, and a marked side.
- For $\triangle QPR$: We have one marked angle at $Q$ (congruent to the angle at $H$ or $Y$), one marked angle at $P$, and a marked side. Wait, let's re - examine the triangles. Looking at $\triangle FHG$ and $\triangle QPR$:
- $\triangle FHG$: Angles at $H$ (marked with two arcs) and $G$ (marked with one arc), and side $FH$ (or the marked side).
- $\triangle QPR$: Angles at $Q$ (marked with two arcs) and $P$ (marked with one arc), and the marked side. Also, looking at $\triangle YXZ$: Triangle $YXZ$ has angle at $Y$ (two arcs), angle at $Z$ (one arc), and marked side $YX$.
- Wait, actually, $\triangle FHG$ and $\triangle QPR$: The angle with two arcs (at $H$ in $\triangle FHG$ and at $Q$ in $\triangle QPR$) are congruent, the angle with one arc (at $G$ in $\triangle FHG$ and at $P$ in $\triangle QPR$) are congruent, and the non - included side (the marked side) is congruent. Also, $\triangle FHG$ and $\triangle QPR$: Let's check the correspondence.
- Alternatively, $\triangle FHG$ and $\triangle QPR$: $\angle H\cong\angle Q$ (two - arc angles), $\angle G\cong\angle P$ (one - arc angles), and the side between the non - included (the marked side) is congruent. Wait, another way: $\triangle FHG$ and $\triangle QPR$:
- $\angle H=\angle Q$ (congruent angle marks), $\angle G = \angle P$ (congruent angle marks), and the side $FH$ (or the marked side) in $\triangle FHG$ corresponds to side $QR$ (or the marked side) in $\triangle QPR$? No, wait, let's look at the triangle $\triangle FHG$ and $\triangle QPR$:
- $\triangle FHG$: Vertices $F$, $H$, $G$; $\triangle QPR$: Vertices $Q$, $P$, $R$.
- The angle at $H$ (in $\triangle FHG$) and angle at $Q$ (in $\triangle QPR$) are congruent (two - arc marks), angle at $G$ (in $\triangle FHG$) and angle at $P$ (in $\triangle QPR$) are congruent (one - arc marks), and the side $FH$ (marked) in $\triangle FHG$ and the marked side in $\triangle QPR$ (the side with the tick mark) are congruent.
- Wait, actually, the correct correspondence is $\triangle FHG\cong\triangle QPR$? No, wait, let's re - check the triangles. Wait, the triangle $\triangle FHG$: angles at $H$ (two arcs), $G$ (one arc), side $FG$ (marked). Triangle $\triangle QPR$: angles at $Q$ (two arcs), $P$ (one arc), side $QR$ (marked). Also, the other triangle is $\triangle YXZ$: angles at $Y$ (two arcs), $Z$ (one arc), side $YX$ (marked).
- Wait, maybe I made a mistake. Let's start over. The AAS requires two angles and a non - included side. So, for $\triangle FHG$: we have $\angle H$ (two arcs), $\angle G$ (one arc), and side $FH$ (the side not between them, or the marked side). For $\triangle QPR$: $\angle Q$ (two arcs), $\angle P$ (one arc), and the marked side (not between the two angles). So the congruence should be $\triangle FHG\cong\triangle QPR$? Wait, no, looking at the labels:
- Wait, the triangle on the left is $\triangle FHG$, middle is $\triangle YXZ$, right is $\triangle QPR$.
- $\triangle FHG$: angles at $H$ (two arcs), $G$ (one arc), side $FH$ (marked).
- $\triangle QPR$: angles at $Q$ (two arcs), $P$ (one arc), side $QR$ (marked).
- $\triangle YXZ$: angles at $Y$ (two arcs), $Z$ (one arc), side $YX$ (marked).
- Now, $\triangle FHG$ and $\triangle QPR$: $\angle H\cong\angle Q$ (two - arc angles), $\angle G\cong\angle P$ (one - arc angles), and the non - included side (the marked side) is congruent. So the congruence statement is $\triangle FHG\cong\triangle QPR$? Wait, no, maybe $\triangle FHG\cong\triangle QPR$? Wait, let's check the vertex correspondence.
- Alternatively, $\triangle FHG$ and $\triangle QPR$: Let's see the angles. The angle with two arcs: in $\triangle FHG$ it's at $H$, in $\triangle QPR$ it's at $Q$. The angle with one arc: in $\triangle FHG$ it's at $G$, in $\triangle QPR$ it's at $P$. The side: the marked side in $\triangle FHG$ (between $F$ and $G$? No, the marked side is on the side from $F$ to the side with the tick mark. Wait, maybe the correct correspondence is $\triangle FHG\cong\triangle QPR$. Wait, another approach: The triangle $\triangle FHG$ and $\triangle QPR$:
- $\angle H\cong\angle Q$ (congruent two - arc angles)
- $\angle G\cong\angle P$ (congruent one - arc angles)
- The side opposite to the one - arc angle (or the non - included side) is congruent (the marked side). So by AAS, $\triangle FHG\cong\triangle QPR$.

Answer:

$\triangle FHG\cong\triangle QPR$