Question

$\overline{ux}\cong\overline{tu},\overline{vw}\cong\overline{ty}$, and $\overline{xy}\cong\overline{uv}$. complete the proof that $\angle y\cong\angle v$.

Explanation:

Step1: Given

\(\overline{UX}\cong\overline{TU}\), \(\overline{VW}\cong\overline{TY}\), \(\overline{XY}\cong\overline{UV}\) (Given)

Step2: Segment addition postulate

\(UY = XY+UX\) (Segment Addition Postulate)

Step3: Segment addition postulate

\(VX=UV + UX\) (Segment Addition Postulate)

Step4: Substitution

Since \(\overline{XY}\cong\overline{UV}\), then \(UY=UV + UX\) (Substitution)

Step5: Transitive property

\(VX = UY\) (Transitive Property of Equality)

Step6: SSS Congruence

\(\triangle T U Y\cong\triangle WXV\) (SSS Congruence: \(\overline{VW}\cong\overline{TY}\), \(VX = UY\), \(\overline{UX}\cong\overline{TU}\))

Step7: CPCTC

\(\angle Y\cong\angle V\) (CPCTC - Corresponding Parts of Congruent Triangles are Congruent)

Answer:

1. Given
2. Given
3. Given
4. Segment Addition Postulate
5. Segment Addition Postulate
6. Substitution (because \(XY = UV\))
7. Transitive Property of Equality
8. SSS (\(\overline{VW}\cong\overline{TY}\), \(VX = UY\), \(\overline{UX}\cong\overline{TU}\))
9. CPCTC