Question

question
given: (abcd) is a parallelogram and (overline{ce} cong overline{fa}).
prove: (fbed) is a parallelogram.

step	statement	reason
2	(overline{bc} cong overline{ad})	opposite sides of a parallelogram / rhombus / rectangle / square are congruent
3	(overline{be} cong overline{fd})	congruent segments added to congruent segments form congruent segments
4	(overline{bc} parallel overline{ad})	select a reason.

Explanation:

Step 1: Recall parallelogram property

In a parallelogram, opposite sides are parallel. Since \(ABCD\) is a parallelogram (given in step 1), by the definition of a parallelogram, opposite sides \(\overline{BC}\) and \(\overline{AD}\) must be parallel. So the reason for \(\overline{BC} \parallel \overline{AD}\) is "Opposite sides of a parallelogram are parallel".

Step 2: Continue proving \(FBED\) is a parallelogram (optional, but to complete the proof idea)

We already have \(\overline{BE} \cong \overline{FD}\) (from step 3) and now we know \(\overline{BC} \parallel \overline{AD}\), which implies \(\overline{BE} \parallel \overline{FD}\) (since \(BE\) is an extension of \(BC\) and \(FD\) is an extension of \(AD\) in the diagram context). A quadrilateral with one pair of opposite sides both congruent and parallel is a parallelogram, so \(FBED\) is a parallelogram. But for step 4's reason, we focus on the parallel side property of the parallelogram.

Answer:

The reason for \(\overline{BC} \parallel \overline{AD}\) is "Opposite sides of a parallelogram are parallel". To fully prove \(FBED\) is a parallelogram, since \(\overline{BE} \cong \overline{FD}\) (from step 3) and \(\overline{BE} \parallel \overline{FD}\) (from the parallelogram \(ABCD\)'s opposite sides parallel property), by the theorem "If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram", \(FBED\) is a parallelogram. But for step 4's reason, the answer to the reason for \(\overline{BC} \parallel \overline{AD}\) is "Opposite sides of a parallelogram are parallel".