Question

complete the proof that $\triangle wxy \cong \triangle vuz$.

1. $\angle vuz \cong \angle wxy$ \qquad given
2. $\angle uzv \cong \angle wyx$ \qquad given
3. $\overline{yz} \cong \overline{vw}$ \qquad given
4. $vz = yz + vy$ \qquad \underline{\hspace{5cm}}
5. $wy = vw + vy$ \qquad additive property of length
6. $vz = vw + vy$ \qquad \underline{\hspace{5cm}}
7. $wy = vz$ \qquad transitive property of equality
8. $\triangle wxy \cong \triangle vuz$ \qquad \underline{\hspace{5cm}}

Explanation:

Step1: Identify reason for Step4

The statement $VZ = YZ + VY$ uses the fact that a segment is equal to the sum of its adjacent sub-segments.
Reason: Additive Property of Length

Step2: Identify reason for Step6

We substitute $\overline{YZ} \cong \overline{VW}$ (so $YZ = VW$) into the equation from Step4.
Reason: Substitution Property of Equality

Step3: Identify reason for Step8

We have two pairs of congruent angles ($\angle VUZ \cong \angle WXY$, $\angle UZV \cong \angle WYX$) and the included side between them congruent ($WY = VZ$), so we use the Angle-Side-Angle congruence rule.
Reason: ASA (Angle-Side-Angle) Congruence Postulate

Answer:

1. Reason for Statement 4: Additive Property of Length
2. Reason for Statement 6: Substitution Property of Equality
3. Reason for Statement 8: ASA (Angle-Side-Angle) Congruence Postulate

Filled proof table:

Statement	Reason
2. $\angle UZV \cong \angle WYX$	Given
3. $\overline{YZ} \cong \overline{VW}$	Given
4. $VZ = YZ + VY$	Additive Property of Length
5. $WY = VW + VY$	Additive Property of Length
6. $VZ = VW + VY$	Substitution Property of Equality
7. $WY = VZ$	Transitive Property of Equality
8. $\Delta WXY \cong \Delta VUZ$	ASA (Angle-Side-Angle) Congruence Postulate