Question

ux bisects ∠vuw. complete the proof that vx ≅ wx.
(image of a triangle with vertices u, v, w, and point x on vw. uw and uv are marked congruent. a table with statements and reasons: 1. ux bisects ∠vuw (given), 2. uw ≅ uv (given), 3. ∠vux ≅ ∠wux, 4. ux ≅ ux, 5. △uvx ≅ △uwx, 6. vx ≅ wx (cpctc))

Explanation:

Step1: Define angle bisector property

An angle bisector splits an angle into two congruent angles. So, since $\overrightarrow{UX}$ bisects $\angle VUW$, $\angle VUX \cong \angle WUX$. Reason: Definition of angle bisector

Step2: Identify reflexive property

A segment is congruent to itself. $\overline{UX} \cong \overline{UX}$ by the Reflexive Property of Congruence. Reason: Reflexive Property of Congruence

Step3: Prove triangle congruence

We have $\overline{UW} \cong \overline{UV}$ (given), $\angle VUX \cong \angle WUX$ (from step1), and $\overline{UX} \cong \overline{UX}$ (from step2). This satisfies the SAS (Side-Angle-Side) Congruence Postulate. So $\triangle UVX \cong \triangle UWX$. Reason: SAS Congruence Postulate

Answer:

Statement	Reason
2. $\overline{UW} \cong \overline{UV}$	Given
3. $\angle VUX \cong \angle WUX$	Definition of angle bisector
4. $\overline{UX} \cong \overline{UX}$	Reflexive Property of Congruence
5. $\triangle UVX \cong \triangle UWX$	SAS Congruence Postulate
6. $\overline{VX} \cong \overline{WX}$	CPCTC