Question

statement reason

1. \\( \overline{w y} \\) and \\( \overline{x z} \\) bisect each other. given
2. \\( w a=y a \\) \\( x a=z a \\) definition of bisection
3. \\( m \angle y a z=m \angle x a w \\) vertical angles theorem
4. sas criterion for congruence
5. \\( \angle y w x \cong \angle w y z \\) corresponding angles of congruent triangles are congruent
6. \\( \overline{y z} \\| \overline{x w} \\) converse of alternate interior angles theorem
7. \\( m \angle x a y=m \angle z a w \\) vertical angles theorem
8. \\( \triangle x a y \cong \triangle z a w \\) sas criterion for congruence
9. \\( \angle x z w \cong \angle z x y \\) corresponding angles of congruent triangles are congruent.
10. \\( \overline{x y} \\| \overline{w z} \\) converse of alternate interior angles theorem
11. quadrilateral wxyz is a parallelogram definition of parallelogram

what is the missing statement in this proof?
a. \\( \triangle x w y \cong \triangle z y w \\)
b. \\( \triangle y a z \cong \triangle w a x \\)
c. \\( w y=x z \\)
d. \\( w y \\| x z \\)

Explanation:

Step1: Recall SAS congruence criteria

SAS (Side - Angle - Side) congruence criteria states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.

Step2: Identify sides and angle

From step 2, we have \(WA = YA\) and \(XA=ZA\). From step 3, \(m\angle YAZ=m\angle XAW\). So the two triangles \(\triangle YAZ\) and \(\triangle WAX\) have two sides and the included angle equal.

Answer:

B. \(\triangle YAZ\cong\triangle WAX\)