Question

uw bisects ∠vux and ∠vwx. complete the proof that ( overline{uv} cong overline{ux} ).

1. ( overleftrightarrow{uw} ) bisects ( angle vux ) | given
2. ( overleftrightarrow{uw} ) bisects ( angle vwx ) | given
3. ( angle vuw cong angle wux ) | definition of angle bisector
4. ( angle uwv cong angle uwx ) | definition of angle bisector
5. ( overline{uw} cong overline{uw} ) | reflexive property of congruence
6. ( \triangle uvw cong \triangle uxw ) |
7. ( overline{uv} cong overline{ux} ) |

Explanation:

Step1: Identify congruent angles/sides

We have $\angle VUW \cong \angle WUX$, $\angle UWV \cong \angle UWX$, and $\overline{UW} \cong \overline{UW}$.

Step2: Prove triangle congruence

Two pairs of corresponding angles and the included side are congruent, so use ASA (Angle-Side-Angle) Congruence Criterion for $\triangle UVW \cong \triangle UXW$.

Step3: Prove segment congruence

Corresponding parts of congruent triangles are congruent, so $\overline{UV} \cong \overline{UX}$.

Answer:

Statement	Reason
7. $\overline{UV} \cong \overline{UX}$	Corresponding Parts of Congruent Triangles are Congruent (CPCTC)