Question

statement: 1. \\(\overline{st} \cong \overline{tu}\\) 2. \\(\overline{tx} \cong \overline{tw}\\) 3. \\(\overline{vw} \cong \overline{xy}\\) 4. \\(\overline{sy} \cong \overline{uv}\\) 5. \\(sw = st + tw\\) 6. \\(ux = tu + tx\\) 7. \\(sw = tu + tx\\) 8. \\(sw = ux\\) 9. \\(wy = xy + wx\\) 10. \\(vx = vw + wx\\) 11. \\(wy = vw + wx\\) 12. \\(vx = wy\\) 13. \\(\triangle swy \cong \triangle uxw\\) reason: 1. given 2. given 3. given 4. given 5. additive property of length 6. additive property of length 7. substitution 8. transitive property of equality 9. additive property of length 10. additive property of length 11. substitution 12. substitution 13. substitution

Explanation:

Step1: Verify missing reason for Step12

We already have that $WY = VW + WX$ (Step11) and $VX = VW + WX$ (Step10). By transitive property, $VX=WY$. But wait, looking at the flow: we know $SW=UX$ (Step8), $SW=ST+TW$ (Step5), $UX=TU+TX$ (Step6), and given $\overline{ST}\cong\overline{TU}$, $\overline{TX}\cong\overline{TW}$, so $SW=UX$. Also, $\overline{SY}\cong\overline{UV}$, $\overline{VW}\cong\overline{XY}$. For Step12, we use the earlier derived equal lengths: from Step9 $WY=XY+WX$, Step10 $VX=VW+WX$, and since $\overline{VW}\cong\overline{XY}$ (so $VW=XY$), substituting gives $VX=WY$. The reason here is Transitive Property of Equality (since $WY=XY+WX$, $VX=VW+WX$, $XY=VW$, so $VX=WY$ via transitivity).

Step2: Verify reason for Step13

We have:

1. $\overline{SY}\cong\overline{UV}$ (Given, Step4)
2. $VX=WY$ (Step12) so $\overline{WY}\cong\overline{VX}$
3. $\overline{VW}\cong\overline{XY}$ (Given, Step3)

By Side-Side-Side (SSS) Congruence Postulate, $\triangle SWY \cong \triangle UXV$.

Answer:

1. Reason for Statement 12: Transitive Property of Equality
2. Reason for Statement 13: SSS Congruence Postulate