Question

1. sas criterion for congruence
2. ∠ywx ≅ ∠wyz corresponding angles of congruent triangles are congruent.
3. (overline{yz}paralleloverline{xw}) converse of alternate interior angles theorem
4. (mangle xay = mangle zaw) vertical angles theorem
5. (\triangle xaycong\triangle zaw) sas criterion for congruence
6. ∠xzw ≅ ∠zxy corresponding angles of congruent triangles are congruent.
7. (overline{xy}paralleloverline{wz}) converse of alternate interior angles theorem
8. quadrilateral wxyz is a parallelogram. definition of parallelogram

what is the missing statement in this proof?
○ a. (\triangle xwycong\triangle zyw)
○ b. (\triangle yazcong\triangle wax)
○ c. (wy = xz)
○ d. (overline{wy}paralleloverline{xz})

Explanation:

• For option A: If \(\triangle XWY\cong\triangle ZYW\) by SAS (assuming appropriate side - angle - side relations from the context of the proof, which is likely given earlier in the overall proof structure not shown here completely), then it can be used as a step before using the property that corresponding angles of congruent triangles are congruent (\(\angle YWX\cong\angle WYZ\)).
• Option B: \(\triangle YAZ\) and \(\triangle WAX\) do not seem to be related to the angles \(\angle YWX\) and \(\angle WYZ\) which are used in the subsequent steps about parallel lines (\(\overline{YZ}\parallel\overline{XW}\)).
• Option C: \(WY = XZ\) is a side - equality. But the next step is about angle - congruence (\(\angle YWX\cong\angle WYZ\)), and side - equality alone (without an angle in between for SAS) won't directly lead to the angle - congruence property used.
• Option D: \(\overline{WY}\parallel\overline{XZ}\) is about parallel lines. But the step after the missing statement is about an angle - congruence (\(\angle YWX\cong\angle WYZ\)) which is a property of congruent triangles, not directly from a parallel - line relationship.

Answer:

A. \(\triangle XWY\cong\triangle ZYW\)