Question

complete the proof that ( overline{yz} cong overline{wz} ).

statement	reason
2	( overline{vy} cong overline{wx} )	given
3	( overline{vy} perp overline{vx} )	given
4	( angle v cong angle x )	all right angles are congruent
5	( angle vzy cong angle wzx )
6	( \triangle vyz cong \triangle xwz )
7	( overline{yz} cong overline{wz} )

Explanation:

Step1: Vertical angles are congruent

$\angle VZY\cong\angle WZX$ (Vertical angles are congruent)

Step2: AAS (Angle - Angle - Side) congruence

In $\triangle VYZ$ and $\triangle XWZ$, we have $\angle V\cong\angle X$ (from step 4), $\angle VZY\cong\angle WZX$ (from step 5), and $\overline{VY}\cong\overline{WX}$ (from step 2). So, $\triangle VYZ\cong\triangle XWZ$ (AAS congruence)

Step3: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since $\triangle VYZ\cong\triangle XWZ$, then $\overline{YZ}\cong\overline{WZ}$ (CPCTC)

Answer:

1. Vertical angles are congruent; 6. AAS (Angle - Angle - Side) Congruence; 7. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)