Which congruence theorem can be used to prove △WXZ ≅ △YZX?

Question

Triangles W X Z and Y Z X share side X Z. Angles W X Z and X Z Y are right angles. Angles X W Z and X Y Z are congruent.

AAS
ASA
SAS
HL

Answer 1

To determine which congruence theorem can be used to prove that triangles $\mathrm{△}WXZ$ and $\mathrm{△}YZX$ are congruent, we can analyze the given information:

1. The triangles share side $XZ$.
2. Angles $WXZ$ and $XZY$ are right angles (thus, they are both ${90}^{\circ }$).
3. Angles $XWZ$ and $XYZ$ are stated to be congruent.

Given these pieces of information:

• Since $WXZ$ and $XZY$ are both right angles, we have: $$\mathrm{\angle }WXZ\cong \mathrm{\angle }YXZ(bothrightangles)$$

• We also have: $$\mathrm{\angle }XWZ\cong \mathrm{\angle }XYZ(given)$$

In triangle $\mathrm{△}WXZ$, 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 $\mathrm{△}WXZ\cong \mathrm{△}YZX$ is ASA (Angle-Side-Angle).