To determine which congruence theorem can be used to prove that triangles \( \triangle WXZ \) and \( \triangle YZX \) are congruent, we can analyze the given information:
- The triangles share side \( XZ \).
- Angles \( WXZ \) and \( XZY \) are right angles (thus, they are both \( 90^\circ \)).
- Angles \( XWZ \) and \( XYZ \) are stated to be congruent.
Given these pieces of information:
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Since \( WXZ \) and \( XZY \) are both right angles, we have: \[ \angle WXZ \cong \angle YXZ \quad (both \ right \ angles ) \]
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We also have: \[ \angle XWZ \cong \angle XYZ \quad (given) \]
In triangle \( \triangle WXZ \), we have:
- Angle \( WXZ \) (which is \( 90^\circ \)),
- Angle \( XWZ \) (which is congruent to angle \( XYZ \)),
- The shared side \( XZ \).
From these, we can conclude that we have two angles and the included side (the right angle acts as the included side with respect to the other sides). This corresponds to the Angle-Side-Angle (ASA) postulate.
Therefore, the congruence theorem that can be used to prove \( \triangle WXZ \cong \triangle YZX \) is ASA (Angle-Side-Angle).