Question

$\angle vxw \cong \angle xvy$ and $\angle wvx \cong \angle vxy$. complete the proof that $\triangle vwx \cong \triangle xyv$.

(image of a quadrilateral divided into two triangles by a diagonal $vx$, with vertices labeled $w$, $x$, $v$, $y$)

statement	reason
2. $\angle wvx \cong \angle vxy$	given
3. $\overline{vx} \cong \overline{vx}$
4. $\triangle vwx \cong \triangle xyv$

Explanation:

Step1: Analyze Statement 3

The segment \( \overline{VX} \) is common to both triangles \( \triangle VWX \) and \( \triangle XYV \). So, \( \overline{VX} \cong \overline{VX} \) by the Reflexive Property of Congruence (a segment is congruent to itself).

Step2: Determine Triangle Congruence

We have two pairs of congruent angles (\( \angle VXW \cong \angle XVY \), \( \angle WVX \cong \angle VXY \)) and a common side (\( \overline{VX} \cong \overline{VX} \)). By the Angle - Angle - Side (AAS) Congruence Criterion, \( \triangle VWX \cong \triangle XYV \).

For Statement 3, the reason is "Reflexive Property of Congruence (a segment is congruent to itself)". For Statement 4, the reason is "AAS (Angle - Angle - Side) Congruence Criterion".

Answer:

• For Statement 3 Reason: Reflexive Property of Congruence (a segment is congruent to itself)
• For Statement 4 Reason: AAS (Angle - Angle - Side) Congruence Criterion