Question

$$ \overleftrightarrow{v y} $$ bisects $$ \angle w v x $$. complete the proof that $$ \triangle v x y \cong \triangle v w y $$.

statement	reason
2. $$ \overline{v w} \cong \overline{v x} $$
3. $$ \angle w v y \cong \angle x v y $$
4. $$ \overline{v y} \cong \overline{v y} $$
5. $$ \triangle v x y \cong \triangle v w y $$

Explanation:

Step1: Given

Given that \( \overrightarrow{VY} \) bisects \( \angle WVX \).

Step2: Given

From the figure, \( \overline{VW}\cong\overline{VX} \) (marked as equal in the diagram).

Step3: Definition of angle - bisector

Since \( \overrightarrow{VY} \) bisects \( \angle WVX \), by the definition of an angle - bisector, \( \angle WVY=\angle XVY \).

Step4: Reflexive property

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

Step5: SAS (Side - Angle - Side) congruence criterion

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. Here, in \( \triangle VXY \) and \( \triangle VWY \), we have \( \overline{VW}\cong\overline{VX} \), \( \angle WVY\cong\angle XVY \), and \( \overline{VY}\cong\overline{VY} \). So, \( \triangle VXY\cong\triangle VWY \) by the SAS (Side - Angle - Side) congruence criterion.

Answer:

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