Question

complete the proof that \\(\overline{uz} \cong \overline{wx}\\).
image of two triangles with labeled points u, v, z and w, v, x, y, z, and a table with statements and reasons

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

Explanation:

Step1: Identify given congruent angles

$\angle WXY \cong \angle VUZ$ (Given, marked angles)
$\angle UZV \cong \angle XWY$ (Given, marked angles)

Step2: Identify equal segments

$\overline{VW} \cong \overline{YZ}$ (Given, marked segments)

Step3: Segment addition postulate

$VZ = YZ + VY$ (Segment Addition Postulate: $V-Y-Z$ colinear)
$WY = VW + VY$ (Segment Addition Postulate: $V-W-Y$ colinear)

Step4: Substitute congruent segments

$VZ = VW + VY$ (Substitute $\overline{VW} \cong \overline{YZ}$ into $VZ = YZ + VY$)

Step5: Prove $WY = VZ$

$WY = VZ$ (Transitive Property: $WY = VW + VY$ and $VZ = VW + VY$)

Step6: Prove triangle congruence

$\triangle UVZ \cong \triangle XYW$ (AAS: $\angle UZV \cong \angle XWY$, $\angle VUZ \cong \angle WXY$, $\overline{VZ} \cong \overline{WY}$)

Step7: Corresponding parts of congruent triangles

$\overline{UZ} \cong \overline{WX}$ (CPCTC: Corresponding Parts of Congruent Triangles are Congruent)

Answer:

$\overline{UZ} \cong \overline{WX}$ is proven via AAS congruence of $\triangle UVZ$ and $\triangle XYW$, then CPCTC.