##### Parallelogram

Parallelogram is a four sided shape where opposite sides are parallel.

Properties:

1. Opposite sides are

2. Opposite angles are congruent.

3. Consecutive angles are supplementary.

4. The diagonals of a parallelogram bisect each other.

5. Each diagonal of a parallelogram separates it into two congruent triangles.

6. If one angle is right, then all the angles are right.

### The Area and Perimeter of Parallelogram

1. The area of parallelogram is measurement of the surface of a shape.

The area of is found by multiplying the base with the height.

The distance between two opposite sides on a parallelogram is called the height.

Area = base x height

$\displaystyle A=b\cdot h$

The perimeter of a parallelogram is the total length of the sides.

Perimeter is  found by adding the lengths of all sides. In this case the perimeter is:

Perimeter = 2 times the(base + side lengths)

$\displaystyle P=2\left( B+S \right)$

Example 1: On the figure below the diagonals of the parallelogram are $\displaystyle AC=10cm$, $\displaystyle BD=14cm$ and the side $\displaystyle AD=6cm$. Find the perimeter of the triangle $\displaystyle BOC$ where $\displaystyle O$ is the intersection point of diagonals.

Solution: From the properties of parallelogram we know that the diagonals bisect each other into equal parts.

So we get  $\displaystyle AO=OC=5cm$ and $\displaystyle DO=OB=7cm$.

Again from the properties we know that opposite sides are congruent.

So we get $\displaystyle AD=BC=6cm$.

Now we know all the sides of the triangle $\displaystyle BOC$ to find the perimeter $\displaystyle P=OB+OC+BC=$ $latex \displaystyle 7+8+6=18cm$.

Example 2: On the figure below the sides of the parallelogram are $\displaystyle AB=12dm$ and $\displaystyle AD=8dm$. The height over the base AB is $\displaystyle DH=6\text{ }dm$. Find the height over the base BC?

Solution: We built the height $\displaystyle DH\bot AB$ and $\displaystyle DE\bot BC$

We know that the area is found by multiplying base with height.

$\displaystyle Area=AB\cdot DH=BC\cdot DE$ also Area=BC·DE

So Area= AB·DH= BC·DE  from where the height  $\displaystyle DE=\frac{{AB\cdot DH}}{{BC}}=\frac{{12\cdot 6}}{8}=9dm$

Example 3: On the figure below we have $\displaystyle DH\bot AB$, $\displaystyle AH=3cm$$\displaystyle HB=2cm$Find the area.

Solution

We have $\displaystyle AB=AH+HB=3+2=5cm$ and $\displaystyle AD=5cm$.

On the triangle $\displaystyle AHD$ we find $\displaystyle DH$ by using Pythagorean Theorem

$\displaystyle DH=\sqrt{{A{{D}^{2}}-A{{H}^{2}}}}=$ $\displaystyle \sqrt{{25-9}}=4cm$

Then we find the area since we know the base and we found the height.

$\displaystyle Area=AB\cdot DH=5\cdot 4=20c{{m}^{2}}$

Example 4: On the figure below we have given three sides of the parallelogram. Find $\displaystyle BC$.

Solution: From the properties of parallelogram we know that the opposite sides are congruent, so we write:

$\displaystyle AB=DC$

$\displaystyle 7x+2=10x-4$

$\displaystyle 10x-7x=2+4$

$\displaystyle 3x=6$

$\displaystyle x=2$

We substitute x to the side $\displaystyle AD$ to find the length $\displaystyle AD=\frac{1}{2}\cdot (2)+1=2$.

From the properties since the opposite sides are congruent then $\displaystyle AD=BC=2$

Example 5: On the figure below we have given the angles of the parallelogram $\displaystyle \widehat{A}={{(10x+2)}^{\circ }}$ and $\displaystyle \widehat{D}={{(7x+8)}^{\circ }}$. Find the angle $\displaystyle \widehat{C}=?$

Solution: From the properties of parallelogram we know that consecutive angles are supplementary so we write:

$\displaystyle \widehat{A}$ is supplementary to $\displaystyle \widehat{D}$

$\displaystyle (10x+2)+(7x+8)=180$

$\displaystyle 10x+7x+2+8=180$

$\displaystyle 17x+10=180$

$\displaystyle 17x=180-10$

$\displaystyle 17x=170$

$\displaystyle x=10$

Based on another property we know that opposite angles are congruent so:

$\displaystyle \widehat{D}=\widehat{C}=7x+8=7\cdot 10+8={{78}^{\circ }}$

Example 6: The perimeter of a parallelogram is $\displaystyle 200cm$. If one of the sides is longer than the other by $\displaystyle 20cm$. Find the length of each side.

Solution: Firstly lets note one of the sides with $\displaystyle x$

Based on the exercise the other side will be $\displaystyle x+20$

We know that the perimeter is found by adding the lengths of all sides.

From the properties the opposite  sides of a parallelogram are congruent so:

$\displaystyle P=x+x+(x+20)+(x+20)=200$

$\displaystyle 4x+40=200$

$\displaystyle 4x=160$

$\displaystyle x=40$

We found that one side is $\displaystyle 40cm$ and the other will be $\displaystyle 40+20=60cm$. Since the opposite sides are congruent then we have to sides of $\displaystyle 40cm$ and two sides of $\displaystyle 60cm$.