Question

$$\overline{tw}\cong\overline{tx},\overline{uv}\cong\overline{sy},\overline{xy}\cong\overline{vw},$$ and $$\overline{tu}\cong\overline{st}$$. complete the proof that $$\angle y\cong\angle v$$.

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

Explanation:

Step1: Substitute \( XY\) with \( VW\)

Since \( XY\cong VW\) (given), then \( WY = VW+WX\) (from step 11, substitution)

Step2: Use Transitive Property of Equality

From step 10 \( VX=VW + WX\), so \( WY=VX\) (transitive property of equality)

Step3: Consider Triangles \( \triangle SYW\) and \( \triangle UVX\)

We know \( \overline{SY}\cong\overline{UV}\) (given), \( \overline{SW}\cong\overline{UX}\) (from step 8, \( SW = UX\)), \( \overline{WY}\cong\overline{VX}\) (from step 2)

Step4: Apply SSS (Side - Side - Side) Congruence Theorem

By SSS, \( \triangle SYW\cong\triangle UVX\)

Step5: Use Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

Since \( \triangle SYW\cong\triangle UVX\), then \( \angle Y\cong\angle V\)

Answer:

1. \( WY = VX\) (Reason: Substitution, using \( XY\cong VW\)); 13. \( \triangle SYW\cong\triangle UVX\) (Reason: SSS Congruence Theorem, since \( \overline{SY}\cong\overline{UV}\), \( \overline{SW}\cong\overline{UX}\), \( \overline{WY}\cong\overline{VX}\)); 14. \( \angle Y\cong\angle V\) (Reason: CPCTC)