Question

which two triangles are congruent? complete the congruence statement.

Explanation:

Step1: Analyze Angle and Side Markings

Look at the triangles: $\triangle QSR$, $\triangle IGH$, and $\triangle YXW$.
- $\triangle QSR$: Angle at $Q$, angle at $S$ (marked), side $SR$ (marked).
- $\triangle YXW$: Angle at $Y$, angle at $X$ (marked), side $XW$ (marked).
- $\triangle IGH$: Angle at $I$, angle at $G$ (marked), side $GH$ (marked).

Check for ASA (Angle-Side-Angle) congruence:
- $\triangle QSR$: $\angle Q$, side $SR$, $\angle S$.
- $\triangle YXW$: $\angle Y$, side $XW$, $\angle X$. Wait, no, let's recheck. Wait, $\triangle QSR$: angle at $Q$ (vertex angle), angle at $S$ (right - like marked), side $SR$ (marked). $\triangle YXW$: angle at $Y$ (vertex angle), angle at $X$ (marked same as $S$), side $XW$ (marked same as $SR$). Wait, actually, $\triangle QSR$ and $\triangle YXW$? No, wait, let's check the angles and sides again.

Wait, $\triangle QSR$: angle at $Q$ (let's say $\angle Q$), angle at $S$ (marked with a red arc), side $SR$ (marked with two ticks). $\triangle YXW$: angle at $Y$ (marked with a red arc, same as $\angle Q$), angle at $X$ (marked with a red arc, same as $\angle S$), side $XW$ (marked with two ticks, same as $SR$). Wait, no, maybe $\triangle QSR$ and $\triangle YXW$? Wait, no, let's do ASA properly.

Wait, the correct congruence: Let's see the markings. $\triangle QSR$: angle at $Q$ (small arc), angle at $S$ (large arc), side $SR$ (two ticks). $\triangle YXW$: angle at $Y$ (small arc, same as $\angle Q$), angle at $X$ (large arc, same as $\angle S$), side $XW$ (two ticks, same as $SR$). So by ASA (Angle - Side - Angle) congruence criterion, $\triangle QSR \cong \triangle YXW$? Wait, no, wait the labels. Wait, the triangle $\triangle QSR$: vertices $Q$, $S$, $R$. $\triangle YXW$: vertices $Y$, $X$, $W$. Wait, maybe I got the labels wrong. Wait, let's check the other triangle. Wait, $\triangle QSR$: angle at $Q$ (vertex), angle at $S$ (base angle), side $SR$ (base). $\triangle YXW$: angle at $Y$ (vertex, same as $\angle Q$), angle at $X$ (base angle, same as $\angle S$), side $XW$ (base, same as $SR$). So the congruence statement should be $\triangle QSR \cong \triangle YXW$? Wait, no, maybe $\triangle QSR \cong \triangle YXW$? Wait, let's check the order of the vertices.

Wait, the correct congruence: Let's look at the angles and sides. The triangle with vertices $Q$, $S$, $R$ and the triangle with vertices $Y$, $X$, $W$. So the congruence statement is $\triangle QSR \cong \triangle YXW$? Wait, no, maybe $\triangle QSR \cong \triangle YXW$? Wait, I think the correct congruence is $\triangle QSR \cong \triangle YXW$. Wait, no, let's do it step by step.

1. Identify corresponding angles: The angle at $Q$ in $\triangle QSR$ corresponds to the angle at $Y$ in $\triangle YXW$ (both have a small red arc). The angle at $S$ in $\triangle QSR$ corresponds to the angle at $X$ in $\triangle YXW$ (both have a large red arc). The side between these angles: $SR$ in $\triangle QSR$ and $XW$ in $\triangle YXW$ (both have two tick marks, so they are congruent). By the ASA (Angle - Side - Angle) congruence postulate, if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. So $\triangle QSR \cong \triangle YXW$.

Step2: Write the Congruence Statement

Based on the ASA congruence, the congruence statement is $\triangle QSR \cong \triangle YXW$. Wait, but let's check the labels again. Wait, the triangle $\triangle QSR$: $Q$ is the top vertex, $S$ and $R$ are the base vertices. $\triangle YXW$: $Y$ is the top vertex, $X$ and $W$ are the base vertices. So the order of the vertices should match the corresponding angles and sides. So $\angle Q$ corresponds to $\angle Y$, $\angle S$ corresponds to $\angle X$, and side $SR$ corresponds to side $XW$. So the congruence statement is $\triangle QSR \cong \triangle YXW$.

Answer:

$\triangle QSR \cong \triangle YXW$ (or depending on the exact markings, but based on the angle and side congruences, the congruent triangles are $\triangle QSR$ and $\triangle YXW$)