Question

complete the proof that δwyz ≅ δywx.
(image of a quadrilateral wxyz with right angles at x, z and congruence marks on segments and angles)

	statement	reason
1	\\(\overline{wz} \perp \overline{yz}\\)	given
2	\\(\overline{wx} \perp \overline{xy}\\)	given
3	\\(\angle wyx \cong \angle ywz\\)	given
4	\\(\angle x \cong \angle z\\)	angles forming a linear pair sum to 180°
5	\\(\overline{wy} \cong \overline{wy}\\)	reflexive property of congruence
6	\\(\delta wyz \cong \delta ywx\\)

Explanation:

Step1: Analyze Given Information

We know \( \overline{WZ} \perp \overline{YZ} \), \( \overline{WX} \perp \overline{XY} \), so \( \angle X = 90^\circ \), \( \angle Z = 90^\circ \), hence \( \angle X \cong \angle Z \) (all right angles are congruent, not the given wrong reason). Also, \( \angle WYX \cong \angle YWZ \) (given), and \( \overline{WY} \cong \overline{WY} \) (reflexive).

Step2: Determine Triangle Congruence Criterion

For triangles \( \triangle WYZ \) and \( \triangle YWX \), we have:

• \( \angle Z \cong \angle X \) (right angles)
• \( \angle YWZ \cong \angle WYX \) (given)
• \( \overline{WY} \cong \overline{WY} \) (common side)

This fits the AAS (Angle - Angle - Side) congruence criterion.

Step3: Correct Reason for Step 4

The reason for \( \angle X \cong \angle Z \) is that all right angles are congruent (since \( \overline{WX} \perp \overline{XY} \) and \( \overline{WZ} \perp \overline{YZ} \) make \( \angle X \) and \( \angle Z \) right angles, so \( \angle X = \angle Z = 90^\circ \)).

Step4: Reason for Step 6

Since we have two angles and a non - included side congruent (AAS), \( \triangle WYZ \cong \triangle YWX \) by AAS Congruence Theorem.

Answer:

• For step 4, the correct reason is "All right angles are congruent".
• For step 6, the reason is "AAS (Angle - Angle - Side) Congruence Theorem".