Question

vw || tu, ∠utv ≅ ∠tuv, and tv ≅ uw. complete the proof that tw ≅ uv.

1 vw || tu
2 ∠utv ≅ ∠tuv
3 tv ≅ uw
4 ∠utv ≅ ∠tvw
5 ∠uwv ≅ ∠tuv
6 ∠tuv ≅ ∠tvw
7 ∠uwv ≅ ∠tvw
8 vw ≅ vw
9 △tvw ≅ △uwv
10 tw ≅ uv

Explanation:

Step1: Given

\(\overline{VW}\parallel\overline{TU}\) is given.

Step2: Alternate - interior angles

When two parallel lines (\(\overline{VW}\) and \(\overline{TU}\)) are cut by a transversal (\(\overline{UW}\)), \(\angle UTV\cong\angle TUV\) (alternate - interior angles).

Step3: Given

\(\overline{TV}\cong\overline{UW}\) is given.

Step4: Alternate - interior angles

When two parallel lines (\(\overline{VW}\) and \(\overline{TU}\)) are cut by a transversal (\(\overline{TV}\)), \(\angle UTV\cong\angle TVW\) (alternate - interior angles).

Step5: Alternate - interior angles

When two parallel lines (\(\overline{VW}\) and \(\overline{TU}\)) are cut by a transversal (\(\overline{UW}\)), \(\angle UWV\cong\angle TUV\) (alternate - interior angles).

Step6: Transitive property

Since \(\angle UTV\cong\angle TUV\) (from step 2) and \(\angle UTV\cong\angle TVW\) (from step 4), by the transitive property of congruence, \(\angle TUV\cong\angle TVW\).

Step7: Transitive property

Since \(\angle UWV\cong\angle TUV\) (from step 5) and \(\angle TUV\cong\angle TVW\) (from step 6), by the transitive property of congruence, \(\angle UWV\cong\angle TVW\).

Step8: Reflexive property

For any segment \(a\), \(a\cong a\). So, \(\overline{VW}\cong\overline{VW}\) (reflexive property of congruence).

Step9: ASA (Angle - Side - Angle)

In \(\triangle TVW\) and \(\triangle UWV\), \(\angle TVW\cong\angle UWV\) (from step 7), \(\overline{VW}\cong\overline{VW}\) (from step 8), \(\angle TWV\cong\angle UVW\) (alternate - interior angles as \(\overline{VW}\parallel\overline{TU}\) and using transversal \(\overline{TW}\) and \(\overline{UV}\)). So, \(\triangle TVW\cong\triangle UWV\) by ASA (Angle - Side - Angle) congruence criterion.

Step10: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle TVW\cong\triangle UWV\), then \(\overline{TW}\cong\overline{UV}\) (CPCTC).

Answer:

1. Given; 2. Alternate - interior angles; 3. Given; 4. Alternate - interior angles; 5. Alternate - interior angles; 6. Transitive property; 7. Transitive property; 8. Reflexive property; 9. ASA (Angle - Side - Angle); 10. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)