Question

ux bisects ∠vuw and uw ≅ uv. complete the proof that ∠w ≅ ∠v.
1 ux bisects ∠vuw
2 uw ≅ uv
3 ∠vux ≅ ∠wux
4 ux ≅ ux
5 △uvx ≅ △uwx
6 ∠w ≅ ∠v

Explanation:

Step1: Given

Statement 1 and 2 are given in the problem.

Step2: Definition of angle bisector

Since \( \overleftrightarrow{UX}\) bisects \( \angle VUW\), by the definition of an angle bisector, \( \angle VUX\cong\angle WUX\).

Step3: Reflexive property

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

Step4: SAS (Side - Angle - Side) congruence criterion

We have \( \overline{UW}\cong\overline{UV}\) (given), \( \angle VUX\cong\angle WUX\) (from angle - bisector definition), and \( \overline{UX}\cong\overline{UX}\) (reflexive property). So, \( \triangle UVX\cong\triangle UWX\) by the SAS (Side - Angle - Side) congruence criterion.

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

Since \( \triangle UVX\cong\triangle UWX\), then \( \angle W\cong\angle V\) by CPCTC.

Answer:

1. Given
2. Given
3. Definition of angle bisector
4. Reflexive property of congruence
5. SAS (Side - Angle - Side)
6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)