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



1. Opposite sides are congruent.

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.


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$.

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