Question

$$\overleftrightarrow{ux}$$ bisects $$\angle vuw$$ and $$\overline{uv} \cong \overline{uw}$$. complete the proof that $$\triangle uvx \cong \triangle uwx$$.

	statement	reason
2	$$\overline{uv} \cong \overline{uw}$$	given
3	$$\angle vux \cong \angle wux$$
4	$$\overline{ux} \cong \overline{ux}$$
5	$$\triangle uvx \cong \triangle uwx$$

Explanation:

Step1: Definition of angle bisector

By the definition of an angle bisector, if a ray bisects an angle, it divides the angle into two congruent angles. So, since \(\overleftrightarrow{UX}\) bisects \(\angle VUW\), we have \(\angle VUX\cong\angle WUX\).

Step2: Reflexive property of congruence

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

Step3: SAS (Side - Angle - Side) congruence criterion

The SAS (Side - Angle - Side) congruence criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. In \(\triangle UVX\) and \(\triangle UWX\), we have \(\overline{UV}\cong\overline{UW}\) (given), \(\angle VUX\cong\angle WUX\) (from step 1), and \(\overline{UX}\cong\overline{UX}\) (from step 2). So, \(\triangle UVX\cong\triangle UWX\) by SAS.

Answer:

1. Definition of angle bisector; 4. Reflexive property of congruence; 5. SAS (Side - Angle - Side) congruence criterion.