Given:In quadrilateral WXYZ , and bisect each other at point A. Prove: Quadrilateral WXYZ is a parallelogram.

Statement

Reason

1. and bisect each other. given
2. WA = YA
XA = ZA

definition of bisection

3. m∠YAZ = m∠XAW

Vertical Angles Theorem

4.

SAS criterion for congruence

5. ∠ YWX ≅ ∠WYZ

Corresponding angles of congruent triangles are congruent.

6. ||

converse of Alternate Interior Angles Theorem

7. m∠XAY = m∠ZAW

Vertical Angles Theorem

8. ∆XAY ≅ ∆ZAW

SAS criterion for congruence

9. ∠XZW ≅ ∠ZXY

Corresponding angles of congruent triangles are congruent.

10. ||

converse of Alternate Interior Angles Theorem

11. Quadrilateral WXYZ is a
parallelogram.

definition of parallelogram