Question

z is the midpoint of \\(\overline{xy}\\) and \\(\overline{wz} \perp \overline{xy}\\). complete the proof that \\(\overline{wy} \cong \overline{wx}\\).

image of a triangle with vertices w, x, y and midpoint z on xy, with wz perpendicular to xy

statement	reason
2. \\(\overline{wz} \perp \overline{xy}\\)	given
3. \\(\angle wzx \cong \angle wzy\\)	all right angles are congruent
4. \\(\overline{xz} \cong \overline{yz}\\)
5. \\(\overline{wz} \cong \overline{wz}\\)
6. \\(\triangle wxz \cong \triangle wyz\\)
7. \\(\overline{wy} \cong \overline{wx}\\)	cpctc

Explanation:

Step1: Reason for midpoint segments

Definition of midpoint: A midpoint divides a segment into two congruent parts. So $\overline{XZ} \cong \overline{YZ}$ because Z is the midpoint of $\overline{XY}$.

Step2: Reason for reflexive congruence

Reflexive Property of Congruence: A segment is congruent to itself. So $\overline{WZ} \cong \overline{WZ}$ by this property.

Step3: Reason for triangle congruence

Side-Angle-Side (SAS) Congruence Postulate: We have two pairs of congruent sides ($\overline{XZ} \cong \overline{YZ}$, $\overline{WZ} \cong \overline{WZ}$) and a pair of congruent included right angles ($\angle WZX \cong \angle WZY$). Thus $\triangle WXZ \cong \triangle WYZ$ by SAS.

Answer:

1. Reason: Definition of midpoint
2. Reason: Reflexive Property of Congruence
3. Reason: Side-Angle-Side (SAS) Congruence Postulate