Question

In the figure, and bisect each other. Complete the statements to prove that quadrilateral ABCD is a parallelogram.

Answer 1

Observe the figure below.

    Statement                                                                          Reason

1. AC and BD bisect each other                                             Given

2. AE = EC and BE = ED                                                         Definition of bisection

3. $m \angle AEB = m \angle CED$                       Vertical angle theorem

Vertical angle theorem states " When two lines intersect each other, the vertically opposite angles are always equal".

4. $\Delta ABE \cong \Delta CDE$                         SAS criterion for congruence

5. $m \angle ACD = m \angle CAB$           Corresponding angles of congruent triangles are congruent

6. $AB \parallel CD$                                 Converse of alternate interior angle theorem

7.  $m \angle BEC = m \angle AED$                   Vertical angle theorem

8.  $\Delta BEC \cong \Delta DEA$            SAS criterion for congruence

9.  $BC \parallel AD$                                 Converse of alternate interior angle theorem

As, $\Delta BEC \cong \Delta DEA$, so $m \angle CBD = m \angle BDA$ as corresponding parts of corresponding triangles are equal. As these angles are alternate interior angles, so the lines BC and AD are parallel by  "Converse of alternate interior angle theorem".