Question

ux bisects ∠vxz and ∠y ≅ ∠w. complete the proof that uw ≅ uy.

Explanation:

Step1: Given

Given in the problem statement.

Step2: Given

Given in the problem statement.

Step3: Vertical angles are congruent

Vertical angles have equal measures.

Step4: Definition of angle bisector

Since \( \overrightarrow{UX}\) bisects \( \angle VXZ\), \( \angle UXZ\cong\angle UXV\).

Step5: Angle addition postulate

The measure of an angle formed by two adjacent angles is the sum of their measures.

Step6: Angle addition postulate

The measure of an angle formed by two adjacent angles is the sum of their measures.

Step7: Substitution property

Substitute \(m\angle UXZ\) with \(m\angle UXV\) (from step 4) in step 5.

Step8: Transitive property

From steps 6 and 7, \(m\angle UXW = m\angle UXY\).

Step9: Reflexive property

Any segment is congruent to itself.

Step10: AAS (Angle - Angle - Side) congruence

\(\angle Y\cong\angle W\) (step 2), \(\angle UXY\cong\angle UXW\) (step 8), \( \overline{UX}\cong\overline{UX}\) (step 9).

Step11: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle UXY\cong\triangle U XW\), \( \overline{UW}\cong\overline{UY}\).

Answer:

1. Given
2. Given
3. Vertical angles are congruent
4. Definition of angle bisector
5. Angle addition postulate
6. Angle addition postulate
7. Substitution property
8. Transitive property
9. Reflexive property
10. AAS (Angle - Angle - Side) congruence
11. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)