Question

1 (overline{vy} cong overline{vx}) given
2 (angle vxz cong angle vyw) given
3 (angle wvx cong angle yvz) given
4 (mangle xvz=mangle xvy + mangle yvz) additive property of angle measure
5 (mangle wyy=mangle wvx + mangle xvy) additive property of angle measure
6 (mangle xvz=mangle xvy + mangle wvx) substitution
7 (mangle wyy=mangle xvz)
8 (\triangle vxz cong \triangle vyw) properties of addition, subtraction, multiplication, and division
reflexive property of congruence
9 (overline{vw} cong overline{vz}) reflexive property of equality
substitution
transitive property of congruence
transitive property of equality

Explanation:

Step1: Analyze the given equations

From statement 4: \(m\angle XVZ=m\angle XVY + m\angle YVZ\), statement 5: \(m\angle WYV=m\angle WVX + m\angle XVY\), and statement 3: \(\angle WVX\cong\angle YVZ\) (so \(m\angle WVX = m\angle YVZ\))

Step2: Substitute \(m\angle YVZ\) with \(m\angle WVX\)

In the equation \(m\angle XVZ=m\angle XVY + m\angle YVZ\) (from statement 4), substitute \(m\angle YVZ\) with \(m\angle WVX\) (because of \(\angle WVX\cong\angle YVZ\)). Then \(m\angle XVZ=m\angle XVY + m\angle WVX\) (statement 6).
From statement 5: \(m\angle WYV=m\angle WVX + m\angle XVY\). By the commutative property of addition \(m\angle WVX + m\angle XVY=m\angle XVY + m\angle WVX\)

Step3: Use substitution

Since \(m\angle XVZ=m\angle XVY + m\angle WVX\) (statement 6) and \(m\angle WYV=m\angle XVY + m\angle WVX\) (from statement 5 and commutative property of addition), we can substitute \(m\angle XVY + m\angle WVX\) in both equations. So \(m\angle WYV=m\angle XVZ\)

Answer:

Substitution