Question

y is the midpoint of \\( \overline{wx} \\) and \\( \overline{wx} \perp \overline{vy} \\). complete the proof that \\( \angle w \cong \angle x \\).

statement	reason
2	\\( \overline{wx} \perp \overline{vy} \\)
3	\\( \angle vyw \cong \angle vyx \\)
4	\\( \overline{wy} \cong \overline{xy} \\)
5	\\( \overline{vy} \cong \overline{vy} \\)
6	\\( \triangle vwy \cong \triangle vxy \\)
7	\\( \angle w \cong \angle x \\)

Explanation:

Step1: Given

Statement 1 and 2 are given in the problem.

Step2: Definition of perpendicular lines

Since \( \overline{WX}\perp\overline{VY}\), by the definition of perpendicular lines, \( \angle VYW = \angle VYX=90^{\circ}\), so \( \angle VYW\cong\angle VYX\) (right - angle congruence).

Step3: Definition of midpoint

If \( Y\) is the midpoint of \( \overline{WX}\), then by the definition of a midpoint, \( \overline{WY}\cong\overline{XY}\).

Step4: Reflexive property

For any segment \( \overline{AB}\), \( \overline{AB}\cong\overline{AB}\). So, \( \overline{VY}\cong\overline{VY}\) by the reflexive property of congruence.

Step5: SAS (Side - Angle - Side) congruence criterion

In \( \triangle VWY\) and \( \triangle VXY\), we have \( \overline{WY}\cong\overline{XY}\) (from step 3), \( \angle VYW\cong\angle VYX\) (from step 2), and \( \overline{VY}\cong\overline{VY}\) (from step 4). So, \( \triangle VWY\cong\triangle VXY\) by the SAS (Side - Angle - Side) congruence criterion.

Step6: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \( \triangle VWY\cong\triangle VXY\), then by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \( \angle W\cong\angle X\).

Answer:

1. Given
2. Given
3. Definition of perpendicular lines (right - angle congruence)
4. Definition of midpoint
5. Reflexive property of congruence
6. SAS (Side - Angle - Side) congruence criterion
7. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)