Question

wx || uv and uv ≅ wx. complete the proof that ux ≅ vw.
1 wx || uv
2 uv ≅ wx
3 ∠u wx ≅ ∠vuw
4 uw ≅ uw
5 △uvw ≅ △wxu
6 ux ≅ vw

Explanation:

Step1: Given

Given in the problem statement.

Step2: Given

Given in the problem statement.

Step3: Alternate interior angles

Since \(\overline{WX}\parallel\overline{UV}\), by the alternate - interior angles theorem, \(\angle UWX\cong\angle VUW\).

Step4: Reflexive property

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

Step5: SAS (Side - Angle - Side)

We have \(\overline{UV}\cong\overline{WX}\) (side), \(\angle VUW\cong\angle UWX\) (angle), \(\overline{UW}\cong\overline{UW}\) (side). So \(\triangle UVW\cong\triangle WXU\) by the SAS (Side - Angle - Side) congruence criterion.

Step6: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle UVW\cong\triangle WXU\), then \(\overline{UX}\cong\overline{VW}\) as corresponding parts of congruent triangles.

Answer:

1. Given; 2. Given; 3. Alternate interior angles theorem; 4. Reflexive property of congruence; 5. SAS (Side - Angle - Side); 6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)