Question

$overline{vw} cong overline{xy}$, $angle y cong angle v$, and $angle swy cong angle uxv$. complete the proof that $\triangle uvx cong \triangle syw$.

(image of a geometric figure with points v, u, w, x, y, s, t)

statement	reason
2 $angle y cong angle v$	given
3 $angle swy cong angle uxv$	given
4 $wy = xy + wx$	additive property of length
5 $vx = vw + wx$	additive property of length
6 $wy = vw + wx$
7 $vx = wy$
8 $\triangle uvx cong \triangle syw$

Explanation:

Step1: Substitute congruent segments

From step 1, $\overline{VW} \cong \overline{XY}$, so $VW = XY$. Substitute $VW$ for $XY$ in step 4:
$WY = VW + WX$

Step2: Set equal substituted expressions

From step 5 ($VX = VW + WX$) and step 6 ($WY = VW + WX$), use transitive property of equality:
$VX = WY$

Step3: Apply ASA congruence rule

We have $\angle V \cong \angle Y$ (step 2), $VX = WY$ (step 7), and $\angle UXV \cong \angle SWY$ (step 3). This satisfies Angle-Side-Angle (ASA) congruence for $\triangle UVX$ and $\triangle SYW$.

Answer:

Statement	Reason
2. $\angle Y \cong \angle V$	Given
3. $\angle SWY \cong \angle UXV$	Given
4. $WY = XY + WX$	Additive Property of Length
5. $VX = VW + WX$	Additive Property of Length
6. $WY = VW + WX$	Substitution Property of Equality
7. $VX = WY$	Transitive Property of Equality
8. $\triangle UVX \cong \triangle SYW$	Angle-Side-Angle (ASA) Congruence Postulate