Question

$\overleftrightarrow{tv}$ bisects $\angle utw$ and $\angle uvw$. complete the proof that $\triangle tvw \cong \triangle tvu$.

	statement	reason
2	$\overleftrightarrow{tv}$ bisects $\angle uvw$	given
3	$\angle utv \cong \angle vtw$
4	$\angle tvu \cong \angle tvw$
5	$\overline{tv} \cong \overline{tv}$	reflexive property of congruence
6	$\triangle tvw \cong \triangle tvu$

Explanation:

• Step 3: Since $\overleftrightarrow{TV}$ bisects $\angle UTW$, it splits the angle into two congruent angles, so $\angle UTV \cong \angle VTW$ by the definition of an angle bisector.
• Step 4: Since $\overleftrightarrow{TV}$ bisects $\angle UVW$, it splits the angle into two congruent angles, so $\angle TVU \cong \angle TVW$ by the definition of an angle bisector.
• Step 6: We have two pairs of congruent angles ($\angle UTV \cong \angle VTW$, $\angle TVU \cong \angle TVW$) and a shared congruent side ($\overline{TV} \cong \overline{TV}$). This matches the Angle-Angle-Side (AAS) congruence criterion, proving $\triangle TVW \cong \triangle TVU$.

Answer:

1. Reason for $\angle UTV \cong \angle VTW$: Definition of angle bisector
2. Reason for $\angle TVU \cong \angle TVW$: Definition of angle bisector
3. Reason for $\triangle TVW \cong \triangle TVU$: Angle-Angle-Side (AAS) Congruence