Question

1. for this question, complete part of the proof of theorem 35 by selecting the best reason or statement.

theorem 35. a quadrilateral is a parallelogram if it contains a pair of opposite sides which are parallel and congruent.
given: $overline{ad}$ is congruent to $overline{bc}$ and $overline{ad}paralleloverline{bc}$ prove: $abcd$ is a parallelogram.
statements reason

1. $overline{ad}$ is congruent to $overline{bc}$ 1. given
2. $overline{ad}paralleloverline{bc}$ 2. given
3. 3. alternate interior angles are congruent
4. $overline{bd}$ is congruent to $overline{bd}$ 4.
5. triangle $adb$ is congruent to triangle $cbd$ 5.
6. 6. cpctc
7. $overline{ab}paralleloverline{dc}$ 7.
8. $abcd$ is a parallelogram. 8. definition of a parallelogram.

therefore: a quadrilateral is a parallelogram if it contains a pair of opposite sides which are parallel and congruent./html
the best reason for statement 7 of this proof is

Explanation:

Step1: Recall triangle - congruence results

Since $\triangle ADB\cong\triangle CBD$ (by step 5), corresponding parts of congruent triangles are equal.

Step2: Identify angle - relationships

The congruence of the triangles gives us equal angles. If $\angle ABD\cong\angle CDB$, then by the converse of the alternate - interior angles theorem (if alternate interior angles are congruent, then the lines are parallel), we can conclude that $\overline{AB}\parallel\overline{DC}$.

Answer:

Converse of alternate - interior angles theorem.