Question

1. $overline{wy}$ and $overline{xz}$ bisect each other. given
2. $wa = ya$

$xa = za$ definition of bisection

1. $mangle yaz = mangle xaw$ vertical angles theorem
2. sas criterion for congruence
3. $angle wmx cong angle wyz$ corresponding angles of congruent triangles are congruent.
4. $overline{yz} parallel overline{xw}$ converse of alternate interior angles theorem
5. $mangle xay = mangle zaw$ vertical angles theorem
6. $\triangle xay cong \triangle zaw$ sas criterion for congruence
7. $angle xzw cong angle zxy$ corresponding angles of congruent triangles are congruent.
8. $overline{xy} parallel overline{wz}$ converse of alternate interior angles theorem
9. quadrilateral $wxyz$ is a parallelogram. definition of parallelogram

what is the missing statement in this proof?
a. $\triangle xwy cong \triangle zyw$
b. $\triangle yaz cong \triangle wax$
c. $wy = xz$
d. $overline{wy} parallel overline{xz}$

Explanation:

Step1: Analyze the Proof Structure

We are proving that quadrilateral \( WXYZ \) is a parallelogram. The proof involves triangle congruence and angle relationships. We need to find the missing statement (step 4) that leads to the congruence of triangles (step 8, \( \triangle XAY \cong \triangle ZAW \)) using the SAS criterion.

Step2: Recall SAS Criterion

The SAS (Side - Angle - Side) criterion for triangle congruence requires two sides and the included angle of one triangle to be congruent to two sides and the included angle of another triangle. We know from step 2 (\( WA = YA \)) and step 3 (\( XA = ZA \)). We need the included angle between these sides to be congruent. The included angles are \( \angle XAY \) and \( \angle ZAW \), which are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent, so \( \angle XAY\cong\angle ZAW \). But wait, step 7 says \( m\angle XAY=m\angle ZAW \) (which is the Vertical Angles Theorem). Wait, step 4 should be the statement that gives the angle congruence? No, wait, step 4 is before step 7. Wait, let's re - examine the table:

Looking at the right - hand side (reasons) and left - hand side (statements):

• Step 1: \( \overline{WY} \) and \( \overline{XZ} \) bisect each other (given)
• Step 2: \( WA = YA \) (definition of bisection)
• Step 3: \( XA = ZA \) (definition of bisection)
• Step 4:? (reason: SAS criterion for congruence? No, the reason for step 8 is SAS. So step 4 should be the angle congruence that, along with steps 2 and 3, satisfies SAS. The angle between \( WA, XA \) in \( \triangle XAY \) and between \( YA, ZA \) in \( \triangle ZAW \) is \( \angle XAY \) and \( \angle ZAW \), which are vertical angles. So the missing statement in step 4 should be \( \angle XAY\cong\angle ZAW \) (by Vertical Angles Theorem, but the reason for step 7 is Vertical Angles Theorem, so step 4 is the statement of angle congruence. Wait, no, the options are:

Option A: \( \triangle XWY\cong\triangle ZYW \)

Option B: \( \triangle YAZ\cong\triangle WAX \)

Option C: \( WY = XZ \)

Option D: \( \overline{WY}\parallel\overline{XZ} \)

Wait, let's think again. We have \( WA = YA \), \( XA = ZA \), and we need the included angle. The vertical angles \( \angle XAY \) and \( \angle ZAW \) are congruent. So the triangles \( \triangle YAZ \) and \( \triangle WAX \): \( YA = WA \) (step 2), \( ZA = XA \) (step 3), and \( \angle YAZ \) and \( \angle WAX \) are vertical angles (so congruent). So by SAS, \( \triangle YAZ\cong\triangle WAX \), which is option B.

Answer:

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