Question

△uxy ≅ △wvy. complete the proof that △uvx ≅ △vwu.

statement | reason
--- | ---
1 △uxy ≅ △wvy |
2 ( overline{ux} cong overline{vw} ) |
3 ( overline{uy} cong overline{vy} ) |
4 ( overline{wy} cong overline{xy} ) |
5 ( angle uxy cong angle vwy ) |
6 ( uw = uy + wy ) |
7 ( vx = vy + xy ) |
8 ( uw = vy + xy ) |
9 ( uw = vx ) |

Explanation:

Step1: Given initial congruence

$\triangle UXY \cong \triangle VWY$ (Given)

Step2: Corresponding sides (CPCTC)

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

Step3: Corresponding sides (CPCTC)

$\overline{UY} \cong \overline{VY}$ (CPCTC)

Step4: Corresponding sides (CPCTC)

$\overline{WY} \cong \overline{XY}$ (CPCTC)

Step5: Corresponding angles (CPCTC)

$\angle UXY \cong \angle VWY$ (CPCTC)

Step6: Segment addition postulate (UW)

$UW = UY + WY$ (Segment Addition Postulate)

Step7: Segment addition postulate (VX)

$VX = VY + XY$ (Segment Addition Postulate)

Step8: Substitute congruent segments

$UW = VY + XY$ (Substitute $\overline{UY} \cong \overline{VY}$, $\overline{WY} \cong \overline{XY}$)

Step9: Transitive property of equality

$UW = VX$ (Transitive Property of Equality, from Steps 7 and 8)

Step10: Prove final triangle congruence

$\triangle UVX \cong \triangle VUW$ (SSS Congruence: $\overline{UX} \cong \overline{VW}$, $\overline{UV} \cong \overline{VU}$ (Reflexive Property), $\overline{VX} \cong \overline{UW}$)

Answer:

Statement	Reason
2. $\overline{UX} \cong \overline{VW}$	CPCTC
3. $\overline{UY} \cong \overline{VY}$	CPCTC
4. $\overline{WY} \cong \overline{XY}$	CPCTC
5. $\angle UXY \cong \angle VWY$	CPCTC
6. $UW = UY + WY$	Segment Addition Postulate
7. $VX = VY + XY$	Segment Addition Postulate
8. $UW = VY + XY$	Substitution Property
9. $UW = VX$	Transitive Property of Equality
10. $\triangle UVX \cong \triangle VUW$	SSS Triangle Congruence Postulate

Final proven result: $\boldsymbol{\triangle UVX \cong \triangle VUW}$