Question

$$ \overline { x y } \perp \overline { y z }, \angle v \cong \angle x $$, and $$ \overline { y z } \perp \overline { v z } $$. complete the proof that $$ \overline { v y } \cong \overline { x z } $$.

statement	reason
2 $$ \angle v \cong \angle x $$	given
3 $$ \overline { y z } \perp \overline { v z } $$	given
4 $$ \angle v z y \cong \angle x y z $$	all right angles are congruent
5 $$ \overline { y z } \cong \overline { y z } $$	reflexive property of congruence
6

Explanation:

Step1: Prove triangles congruent

Since \(\overline{XY}\perp\overline{YZ}\), \(\angle XYZ = 90^{\circ}\). Since \(\overline{YZ}\perp\overline{VZ}\), \(\angle VZY=90^{\circ}\). We know \(\angle V\cong\angle X\) (given), \(\angle VZY\cong\angle XYZ\) (all right - angles are congruent) and \(\overline{YZ}\cong\overline{YZ}\) (reflexive property of congruence). By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle VZY\cong\triangle XYZ\).

Step2: Use congruent triangles property

If two triangles are congruent, then their corresponding sides are congruent. In \(\triangle VZY\) and \(\triangle XYZ\), \(\overline{VY}\) and \(\overline{XZ}\) are corresponding sides.

Answer:

\(\triangle VZY\cong\triangle XYZ\) (by AAS)