Given: and bisect each other. prove: quadrilateral abcd is a parallelogram. proof: statement reason 1. and bisect each other. given 2. ae = ec be = ed definition of bisection 3. m∠aeb = m∠ced 4. δabe ≅ δcde sas criterion 5. ∠acd ≅ ∠cab corresponding angles of congruent triangles are congruent. 6. converse of alternate interior angles theorem 7. m∠bec = m∠aed vertical angles theorem 8. δbec δdea sas criterion for congruence 9. dbc ≅ bda corresponding angles of congruent triangles are congruent. 10. converse of alternate interior angles theorem 11. quadrilateral abcd is a parallelogram. definition of a parallelogram 1 what is the reason for step 3 of this proof? a. alternate interior angles theorem b. corresponding angles in congruent triangles are congruent. c. for parallel lines cut by a transversal, corresponding angles are congruent. d. vertical angles theorem e.

To Prove: Quadrilateral ABCD is a parallelogram.

Proof: In Δ ABE and ΔCDE

   1. AE = EC and BE = ED [ Diagonals bisect each other]

   2.∠ AEB = ∠ CED [ vertically opposite angles]

Δ ABE ≅ ΔCDE----------        [SAS]

∠ ACD ≅ ∠CAB   [Corresponding angles of congruent triangles are congruent⇒This statement is untrue ∴ these are alternate interior angles not corresponding angles.]

6. The converse of alternate interior interior angle theorem states that if two parallel lines are cut by a transversal then alternate interior angles are equal.

7. In ΔBEC and ΔAED

∠BEC = ∠AED  [ Vertical Angles Theorem ]

AE = EC and BE = ED [ Diagonals bisect each other]

⇒ ΔBEC≅ ΔDEA  [ SAS criterion for congruence]

9. DBC ≅ BDA  [ Corresponding angles of congruent triangles are congruent⇒This statement is untrue ∴ these are alternate interior angles not corresponding angles.]

As pair of triangles are congruent ∵ quadrilateral ABCD is a parallelogram.

Step 3 is  m∠AEB = m∠CED

These pair of angles are vertically opposite angles of ΔAEB and ΔCED.

 Option [D. Vertical Angles Theorem]  is correct.