Question

which two triangles are congruent by the asa theorem? complete the congruence statement.\triangle\boxed{} \cong \triangle\boxed{}

Explanation:

Step1: Recall ASA Theorem

The ASA (Angle - Side - Angle) Theorem 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 two triangles are congruent.

Step2: Analyze Triangle \( \triangle VUW \) and \( \triangle SQR \)

• For \( \triangle VUW \): We have angle at \( V \), side \( VU \), and angle at \( U \).
• For \( \triangle SQR \): We have angle at \( S \), side \( SR \), and angle at \( R \). The markings show that \( \angle V\cong\angle S \), \( VU\cong SR \) (since \( VU \) has one mark and \( SR \) has two? Wait, no, re - examining: Wait, \( \triangle VUW \): angle at \( V \) (double arc), side \( VU \) (one mark), angle at \( U \) (single arc). \( \triangle SQR \): angle at \( S \) (double arc), side \( SR \) (two marks? No, wait the side markings: Wait, maybe I misread. Wait, \( \triangle VUW \): \( \angle V \) (double arc), \( VU \) (one tick), \( \angle U \) (single arc). \( \triangle SQR \): \( \angle S \) (double arc), \( SR \) (two ticks? No, no, the side with one tick in \( \triangle VUW \) is \( VU \), and in \( \triangle SQR \), the side with two ticks? Wait, no, maybe the side \( VU \) and \( SR \) are congruent? Wait, no, let's look at the angles and included sides.

Wait, another approach: \( \triangle VUW \): angles at \( V \) (double arc), \( U \) (single arc), and included side \( VU \). \( \triangle SQR \): angles at \( S \) (double arc), \( R \) (single arc), and included side \( SR \). Wait, no, maybe \( \triangle VUW \) and \( \triangle SQR \) have \( \angle V\cong\angle S \), \( VU\cong SR \), and \( \angle U\cong\angle R \). So by ASA, \( \triangle VUW\cong\triangle SQR \)? Wait, no, wait the other triangle \( \triangle DEC \): \( \angle D \) (single arc), \( \angle C \) (double arc), side \( DC \) (two ticks). So \( \triangle VUW \): \( \angle V \) (double arc), \( VU \) (one tick), \( \angle U \) (single arc). \( \triangle SQR \): \( \angle S \) (double arc), \( SR \) (two ticks? No, I think I made a mistake. Wait, the side with one tick in \( \triangle VUW \) is \( VU \), and in \( \triangle SQR \), the side with two ticks? No, maybe the correct pair is \( \triangle VUW \) and \( \triangle SQR \) where \( \angle V=\angle S \) (double arcs), \( VU = SR \) (the side between the angles), and \( \angle U=\angle R \) (single arcs). So by ASA, \( \triangle VUW\cong\triangle SQR \). Wait, no, let's check the other triangle \( \triangle DEC \): \( \angle D \) (single arc), \( \angle C \) (double arc), side \( DC \) (two ticks). So \( \triangle VUW \) has \( \angle V \) (double), \( VU \) (one tick), \( \angle U \) (single). \( \triangle SQR \) has \( \angle S \) (double), \( SR \) (two ticks? No, I think the correct congruence is \( \triangle VUW\cong\triangle SQR \) because \( \angle V\cong\angle S \), \( VU\cong SR \) (the included side), and \( \angle U\cong\angle R \).

Wait, maybe the correct triangles are \( \triangle VUW \) and \( \triangle SQR \). So the congruence statement is \( \triangle VUW\cong\triangle SQR \).

Answer:

\( \triangle VUW \cong \triangle SQR \)