Question

$\angle w \cong \angle z$, $\overline{yz} \cong \overline{wx}$, and $\angle txz \cong \angle vyw$. complete the proof that $\triangle vwy \cong \triangle tzx$.

statement	reason
2 $\overline{yz} \cong \overline{wx}$	given
3 $\angle txz \cong \angle vyw$	given
4 $xz = yz + xy$
5 $wy = wx + xy$
6 $xz = wx + xy$
7 $wy = xz$	transitive property of equality
8 $\triangle vwy \cong \triangle tzx$	asa

Explanation:

Step1: Analyze Statement 4

To find the reason for \( XZ = YZ + XY \), we recognize that this is the Segment Addition Postulate, which states that if a point \( Y \) is between \( X \) and \( Z \), then \( XZ = XY + YZ \).

Step2: Analyze Statement 5

For \( WY = WX + XY \), this is also the Segment Addition Postulate, as \( X \) is between \( W \) and \( Y \), so \( WY = WX + XY \).

Step3: Analyze Statement 6

Given \( YZ \cong WX \), their lengths are equal (\( YZ = WX \)). From Statement 4 (\( XZ = YZ + XY \)) and substituting \( YZ \) with \( WX \) (since \( YZ = WX \)), we use the Substitution Property of Equality to get \( XZ = WX + XY \).

Answer:

• Statement 4 Reason: Segment Addition Postulate
• Statement 5 Reason: Segment Addition Postulate
• Statement 6 Reason: Substitution Property of Equality (since \( YZ = WX \) from \( \overline{YZ} \cong \overline{WX} \))