Question

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.

Answer 1

A and D

Explanation:

Answer 2

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.