Question

Point E is the midpoint of side BC of parallelogram ABCD, and point F is the midpoint of side AD Prove that quadrilateral AFCE is a parallelogram.

Answer 1

Quadrilateral is a parallelogram

Explanation:

By definition of opposite sides congruent theorem, properties of congruency (by that I mean parts of congruent lines are congruent) and properties of parallel lines (parts of parallel lines are parallel) quadrilateral AFCE is a parallelogram. Refer to the other responses picture to see what I mean--

Answer 2

See proof below

Explanation:

Theorem: If a quadrilateral has one set of opposite sides which are both congruent and parallel, then it is a parallelogram.

Consider parallelogram ABCD. A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Thus, BC||AD. A parallelogram has 2 sets of opposite sides congruent. Thus, BC=AD.

If E and F are midpoints of sides BC and AD, then halves of these sides are congruent too (CE=FA).

In the quadrilateral AFCE, we have congruent and parallel sides EC and FA. Thus, this quadrilateral is a parallelogram.