aExample Definition: The midsegment of a parallelogram connects the midpoints of opposite sides.
B
b See solution.
C
c See solution.
Practice makes perfect
a We are asked to write a definition for the midsegment of a parallelogram. Let's use the following definition.
The midsegment of a parallelogram connects the midpoints of opposite sides.
Keep in mind that the wording of your definition can vary.
b The midsegment MN of parallelogram ABCD cuts the parallelogram into two quadrilaterals, AMND and MBCN. Let's focus on quadrilateral AMND. We will show that this is a parallelogram.
Let's compare sides AM and DN. We will show that these are parallel and congruent.
AM is Parallel to DN
Quadrilateral ABCD is a parallelogram, so by definition the opposite sides are parallel. Since segments AM and DN are part of opposite sides of ABCD, we know that these are parallel.
AM∥DN
AM is Congruent to DN
Quadrilateral ABCD is a parallelogram, so according to Theorem 6-3 opposite sides are congruent. This means that they have equal lengths.
AB≅DC⟹ AB=DC
Points M and N are midpoints of these opposite sides, so the lengths of AM and DN are half of the lengths of AB and DC. Since AB=DC, this means that AM and DN also have the same length, so they are congruent.
AM=DN ⟹ AM≅DN
Finishing the Proof
We now know that quadrilateral AMND has two opposite sides that are parallel and congruent.
AM∥DNandAM≅DN
According to Theorem 6-12, this means that AMND is a parallelogram. By definition we can now conclude that the other two sides are also parallel.
MN∥AD
This means that the midsegment is parallel to one side of parallelogram ABCD. Since AD is also parallel to the opposite side, BC, Theorem 3-8 guarantees that the midsegment is also parallel to BC.
MN∥BC
This is illustrated on the diagram below.
2
&Given:&& ABCDis a parallelogram
& && Mis the midponint ofAB
& && Nis the midponint ofDC
&Prove:&& MN∥ADandMN∥BC
Proof:
c Let's draw one of the diagonals of parallelogram ABCD and focus on one of the triangles formed.
According to Part B, the midsegment is parallel to side AB, so MO cuts triangle △ ABD parallel to one of its sides. According to the Side-Splitter Theorem, this means that MO divides sides AB and DB proportionally.
BM/MA=BO/OD
Since M is the midpoint of AB, this proves that O is the midpoint of BD, so the midsegment bisects diagonal BD.
According to Theorem 6-6, the diagonals of a parallelogram bisect each other.
This means that O is also the midpoint of diagonal AC, so the midsegment bisects both diagonals.
We can summarize the steps above in a flow proof.
Completed Proof
2
&Given:&& ABCDis a parallelogram
& && Mis the midponint ofAB
& && Nis the midponint ofDC
&Prove:&& MNbisectsACandBD
Proof:
