Question

1. wx ≅ vz, wy ≅ vy, yz ≅ yx
2. wv ≅ vw

3.
4.
5.
6.
7.

1. △vwx ≅ △wvz
2. given
3. reflexive property of congruence
4. definition of congruent segments
5. segment addition postulate
6. substitution property of equality
7. transitive property of equality
8. definition of congruent segments
9. sss congruence theorem

vx = vy + yx, wz = wy + yz
vx = wz
vx ≅ wz
vx = wy + yz
yz = yx, wy = vy

Explanation:

Step1: Segment Addition Postulate

By the Segment Addition Postulate, \(VX = VY+YX\) and \(WZ = WY + YZ\).

Step2: Substitution

Since \(WX\cong VZ\), \(WY\cong VY\), \(YZ\cong YX\), we substitute \(VY\) for \(WY\) and \(YX\) for \(YZ\) in the equation \(WZ = WY + YZ\). So \(WZ=VY + YX\).

Step3: Transitive Property of Equality

Because \(VX = VY+YX\) and \(WZ=VY + YX\), by the Transitive Property of Equality, \(VX = WZ\).

Step4: Definition of congruent segments

Since \(VX = WZ\), by the Definition of congruent segments, \(\overline{VX}\cong\overline{WZ}\).

Step5: SSS Congruence Theorem

In \(\triangle VWX\) and \(\triangle WVZ\), we have \(WX\cong VZ\), \(WV\cong VW\) (Reflexive Property of Congruence), \(\overline{VX}\cong\overline{WZ}\). So by the SSS (Side - Side - Side) Congruence Theorem, \(\triangle VWX\cong\triangle WVZ\).

Answer:

1. \(VX = VY + YX\), \(WZ=WY + YZ\) (Segment Addition Postulate); 5. \(WZ = VY+YX\) (Substitution Property of Equality); 6. \(VX = WZ\) (Transitive Property of Equality); 7. \(\overline{VX}\cong\overline{WZ}\) (Definition of congruent segments); 8. \(\triangle VWX\cong\triangle WVZ\) (SSS Congruence Theorem)