Question

complete the proof that ( overline{wx} cong overline{uz} ).
1 ( overline{vw} cong overline{yz} )
2 ( angle xwy cong angle uzv )
3 ( angle wxy cong angle vuz )
4 ( vz = yz + vy )
5 ( wy = vw + vy )
6 ( vz = vw + vy )
7 ( wy = vz )
8 ( \triangle uvz cong \triangle xyw )
9 ( overline{wx} cong overline{uz} )

Explanation:

Step1: Analyze the given information

We need to find the reason for \(\overline{VW}\cong\overline{YZ}\). Looking at the figure, we can see that these are given as congruent segments.

Step2: Analyze the reason for \(VZ = YZ+VY\)

This is based on the Additive Property of Length which states that if we have three collinear points \(Y\), \(Z\), \(V\) (or \(V\), \(Y\), \(Z\)) then the length of the whole segment \(VZ\) is the sum of the lengths of its sub - segments \(YZ\) and \(VY\).

Step3: Analyze the reason for \(VZ = VW + VY\)

Since \(\overline{VW}\cong\overline{YZ}\), by the definition of congruent segments \(VW = YZ\). Substituting \(YZ\) with \(VW\) in the equation \(VZ=YZ + VY\) (from step 2) gives \(VZ = VW+VY\) (Substitution Property).

Step4: Analyze the reason for \(WY = VZ\)

From \(WY=VW + VY\) (Additive Property of Length) and \(VZ = VW + VY\) (from step 3), by the Transitive Property of Equality (if \(a=b\) and \(c = b\), then \(a=c\)), we have \(WY = VZ\).

Step5: Analyze the reason for \(\triangle UVZ\cong\triangle XYW\)

We have \(\angle XWY\cong\angle UZV\) (given), \(\angle WXY\cong\angle VUZ\) (given), and \(WY = VZ\) (from step 4). By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle UVZ\cong\triangle XYW\).

Step6: Analyze the reason for \(\overline{WX}\cong\overline{UZ}\)

Since \(\triangle UVZ\cong\triangle XYW\), by the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) property, \(\overline{WX}\cong\overline{UZ}\).

Answer:

1. Given
2. Given
3. Given
4. Additive Property of Length
5. Additive Property of Length
6. Substitution Property
7. Transitive Property of Equality
8. AAS
9. CPCTC