We are asked to show that if we put two parallelograms next to each other, then the resulting quadrilateral is also a parallelogram.
Let's see what we know about segments AF, BE, and CD.
By definition, opposite sides of parallelogram ABEF are parallel, so AF∥BE. Opposite sides of parallelogram BCDE are parallel, so BE∥CD. Since BE is parallel to both AF and CD, these two segments are parallel.
According to Theorem 6.3, opposite sides of parallelogram ABEF are congruent, so AF≅BE. Opposite sides of parallelogram BCDE are congruent, so BE≅CD. Because of the transitive property of congruence, AF≅CD.
The observations above tell us that in quadrilateral ACDF opposite sides AF and CD are both parallel and congruent. According to Theorem 6.12, this means that ACDF is a parallelogram. Let's write a two-column proof as asked.
Completed Proof
Given:& ABEFis a parallelogram
& BCDEis a parallelogram
Prove:& ACDFis a parallelogram
Proof:
Statements
Reasons
1.
ABEF and BCDE are parallelograms.
1.
Given
2.
AF∥BE and BE∥CD
2.
Opposite sides of parallelograms (Definition)
3.
AF≅BE and BE≅CD
3.
Opposite sides of parallelograms (Theorem 6.3)
4.
AF∥CD and AF≅CD
4.
Transitive property
5.
ACDF is a parallelogram.
5.
Opposite sides are congruent and parallel (Theorem 6.12).
