Question

complete the proof that \\(overline{xz} \cong overline{wy}\\).
image of a geometric figure with points w, x, y, z, v and some marked congruent angles and segments

statement	reason
2 \\(angle yvz \cong angle wvx\\)	given
3 \\(angle w \cong angle z\\)	given
4 \\(mangle xvz = mangle xvy + mangle yvz\\)	blank
5 \\(mangle wvy = mangle wvx + mangle xvy\\)	additive property of angle measure
6 \\(mangle xvz = mangle xvy + mangle wvx\\)	blank
7 \\(mangle wvy = mangle xvz\\)	transitive property of equality
8 \\(\triangle vxz \cong \triangle vyw\\)	aas
9 \\(overline{xz} \cong overline{wy}\\)	blank

Explanation:

Step1: Reason for Statement 4

The measure of an angle formed by two adjacent angles is the sum of the measures of those adjacent angles. So for $\angle XVZ$, it is composed of $\angle XVY$ and $\angle YVZ$, so the reason is "Additive Property of Angle Measure".

Step2: Reason for Statement 6

We know from Statement 2 that $\angle YVZ\cong\angle WVX$, which means $m\angle YVZ = m\angle WVX$ (by definition of congruent angles). From Statement 4, $m\angle XVZ=m\angle XVY + m\angle YVZ$. Substituting $m\angle YVZ$ with $m\angle WVX$ (because they are equal), we get $m\angle XVZ=m\angle XVY + m\angle WVX$. The reason here is "Substitution Property (since $m\angle YVZ = m\angle WVX$ from $\angle YVZ\cong\angle WVX$)".

Step3: Reason for Statement 9

If two triangles are congruent (in this case $\triangle VXZ\cong\triangle VYW$ by AAS), then their corresponding sides are congruent. So $\overline{XZ}$ and $\overline{WY}$ are corresponding sides of congruent triangles, so the reason is "Corresponding Parts of Congruent Triangles are Congruent (CPCTC)".

Answer:

• Statement 4 Reason: Additive Property of Angle Measure
• Statement 6 Reason: Substitution (using $\angle YVZ\cong\angle WVX$)
• Statement 9 Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)