Question

For exercises, 14-18 write paragraph proofs.

Given: parallelogram ABCD, M and N are the midpoints of $\overline{\mathrm{AB}}$and $\overline{\mathrm{DC}}$.

Prove: AMCN is a parallelogram.

Short answer

It is being given that ABCD is a parallelogram.

In a parallelogram, both pairs of opposite sides are congruent and parallel.

Therefore, in the parallelogram ABCD, both pairs of opposite sides are congruent and parallel.

Therefore, $\overline{AB}\cong \overline{CD}$, $\overline{\mathrm{AD}}\cong \overline{\mathrm{BC}}$, $\overline{\mathrm{AB}}\parallel \overline{\mathrm{CD}}$and $\overline{\mathrm{AD}}\parallel \overline{\mathrm{BC}}$.

Therefore, $\mathrm{AB}=\mathrm{CD}$.

It is also being given that M and N are the midpoints of $\overline{\mathrm{AB}}$and $\overline{\mathrm{DC}}$.

As, M is the midpoint of $\overline{\mathrm{AB}}$, therefore by using the definition of midpoint it can be said that role="math" localid="1637741738802" $\mathrm{AM}=\frac{1}{2}\mathrm{AB}$.

As, N is the midpoint of $\overline{\mathrm{DC}}$, therefore by using the definition of midpoint it can be said that role="math" localid="1637741802557" $\mathrm{NC}=\frac{1}{2}\mathrm{DC}$.

Therefore, it can be noticed that:

$\begin{array}{c}\mathrm{AB}=\mathrm{CD}\\ \frac{1}{2}\mathbf{AB}\mathbf{=}\frac{1}{2}\mathbf{DC}\\ \mathrm{AM}=\mathrm{NC}\end{array}$

Therefore, $\mathbf{AM}\cong \mathbf{NC}$.

As, $\overline{\mathrm{AB}}\parallel \overline{\mathrm{CD}}$, therefore it can be said that $\overline{\mathrm{AM}}\parallel \overline{\mathrm{NC}}$.

If one pair of opposite side is both congruent and parallel, then the quadrilateral is a parallelogram.

As, $\overline{\mathrm{AM}}\parallel \overline{\mathrm{NC}}$and $\mathbf{AM}\cong \mathbf{NC}$, therefore, the quadrilateral AMCN is a parallelogram.

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Description of step.

It is being given that ABCD is a parallelogram.

In a parallelogram, both pairs of opposite sides are congruent and parallel.

Therefore, in the parallelogram ABCD, both pairs of opposite sides are congruent and parallel.

Therefore, $\overline{AB}\cong \overline{CD},\overline{AD}\cong \overline{BC},\overline{AB}\parallel \overline{CD}$and $\overline{AD}\parallel \overline{BC}$.

Therefore, $AB=CD$.

Step 3. Description of step.

It is also being given that M and N are the midpoints of $\overline{AB}$and $\overline{DC}$.

As, M is the midpoint of $\overline{AB}$, therefore by using the definition of midpoint it can be said that $AM=\frac{1}{2}AB$.

As, N is the midpoint of $\overline{DC}$, therefore by using the definition of midpoint it can be said that $NC=\frac{1}{2}DC$.

Therefore, it can be noticed that:

$\begin{array}{c}AB=CD\\ \frac{1}{2}AB=\frac{1}{2}DC\\ AM=NC\end{array}$

Therefore, $AM\cong NC$.

As, $\overline{AB}\parallel \overline{CD}$, therefore it can be said that $\overline{AM}\parallel \overline{NC}$.

If one pair of the opposite sides is both congruent and parallel, then the quadrilateral is a parallelogram.

As, $\overline{AM}\parallel \overline{NC}$and $AM\cong NC$, therefore, the quadrilateral AMCN is a parallelogram.