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)

reason: (blank spaces for each statements reason)
(there is also a diagram on the right with triangles and points s, y, x, t, w, u)

Explanation:

Step1: Identify given congruent segments

$\overline{ST} \cong \overline{TU}$ (Given)

Step2: Identify given congruent segments

$\overline{TX} \cong \overline{TW}$ (Given)

Step3: Identify given congruent segments

$\overline{VW} \cong \overline{XY}$ (Given)

Step4: Identify given congruent segments

$\overline{SY} \cong \overline{UV}$ (Given)

Step5: Apply segment addition postulate

$SW = ST + TW$ (Segment Addition Postulate: whole = sum of parts)

Step6: Apply segment addition postulate

$UX = TU + TX$ (Segment Addition Postulate: whole = sum of parts)

Step7: Substitute congruent segments (Steps1,2)

$SW = TU + TX$ (Substitute $ST \cong TU$, $TW \cong TX$)

Step8: Transitive property of congruence/equality

$SW = UX$ (Transitive Property: from Steps6,7)

Step9: Apply segment addition postulate

$WY = XY + WX$ (Segment Addition Postulate: whole = sum of parts)

Step10: Apply segment addition postulate

$VX = VW + WX$ (Segment Addition Postulate: whole = sum of parts)

Step11: Substitute congruent segment (Step3)

$WY = VW + WX$ (Substitute $XY \cong VW$)

Step12: Transitive property of equality

$VX = WY$ (Transitive Property: from Steps10,11)

Answer:

Statement	Reason
2. $\overline{TX} \cong \overline{TW}$	Given
3. $\overline{VW} \cong \overline{XY}$	Given
4. $\overline{SY} \cong \overline{UV}$	Given
5. $SW = ST + TW$	Segment Addition Postulate
6. $UX = TU + TX$	Segment Addition Postulate
7. $SW = TU + TX$	Substitution Property (from 1,2,5)
8. $SW = UX$	Transitive Property of Equality
9. $WY = XY + WX$	Segment Addition Postulate
10. $VX = VW + WX$	Segment Addition Postulate
11. $WY = VW + WX$	Substitution Property (from 3,9)
12. $VX = WY$	Transitive Property of Equality