Question

given: ( \triangle bcdcong\triangle efd ), b is the midpoint of ( overline{ac} ). prove: abef is a parallelogram. 7. ( overline{ab}paralleloverline{ef} ) 7. if two lines form congruent alternate interior angles with a transversal, then the lines are parallel. 8. abef is a parallelogram. 8.

Explanation:

Step1: Recall the definition of a parallelogram

A quadrilateral is a parallelogram if one pair of opposite sides is both parallel and congruent.

Step2: Analyze the given information

We are given \( \triangle BCD\cong\triangle EFD\). So, \(BC = EF\) (corresponding parts of congruent triangles). Since \(B\) is the mid - point of \(AC\), \(AB=BC\). Then \(AB = EF\). Also, we have \( \overline{AB}\parallel\overline{EF}\) (given in statement 7 with its reason).

Answer:

A quadrilateral \(ABEF\) has one pair of opposite sides (\(\overline{AB}\) and \(\overline{EF}\)) that are both parallel and congruent. By the definition of a parallelogram (a quadrilateral with one pair of opposite sides parallel and congruent is a parallelogram), \(ABEF\) is a parallelogram. So the reason for statement 8 is "A quadrilateral with one pair of opposite sides parallel and congruent is a parallelogram".