Question

statement reason
1 $overline{vw}congoverline{xy}$ given
2 $overline{st}congoverline{tu}$ given
3 $overline{sy}congoverline{uv}$ given
4 $overline{tx}congoverline{tw}$ given
5 $sw = st+tw$ additive property of length
6 $ux = tu+tx$ additive property of length
7 $sw = tu+tx$ substitution
8 $sw = ux$
9 $wy = xy+wx$ additive property of length
10 $vx = vw+wx$ additive property of length
11 $wy = vw+wx$ substitution
12 $vx = wy$ transitive property of equality
13 $\triangle swycong\triangle uxv$

Explanation:

Step1: Recall given equal - length segments

We are given $\overline{VW}\cong\overline{XY}$, $\overline{ST}\cong\overline{TU}$, $\overline{SY}\cong\overline{UV}$, $\overline{TX}\cong\overline{TW}$.

Step2: Apply additive property of length

By the additive property of length, we have $SW = ST+TW$ (Statement 5) and $UX = TU + TX$ (Statement 6).

Step3: Use substitution

Since $\overline{ST}\cong\overline{TU}$ and $\overline{TX}\cong\overline{TW}$, we substitute in Statement 7: $SW=TU + TX$. And then, because $UX = TU + TX$, we get $SW = UX$ (Statement 8) by the transitive property of equality.

Step4: Apply additive property of length again

We have $WY=XY + WX$ (Statement 9) and $VX = VW+WX$ (Statement 10).

Step5: Use substitution again

Since $\overline{VW}\cong\overline{XY}$, we substitute in Statement 11: $WY = VW+WX$. Then, by the transitive property of equality, $VX = WY$ (Statement 12).

Step6: Prove triangle congruence

We have shown that $SW = UX$, $VX = WY$, and $\overline{SY}\cong\overline{UV}$. By the Side - Side - Side (SSS) congruence criterion, $\triangle SWY\cong\triangle UXV$ (Statement 13).

Answer:

$\triangle SWY\cong\triangle UXV$