# High School Math : Parallelograms

## Example Questions

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### Example Question #328 : Act Math

A parallelogram, with dimensions in cm, is shown below.

What is the perimeter of the parallelogram, in cm?

Explanation:

The triangle on the left side of the figure has a and a  angle. Since all of the angles of a triangle must add up to , we can find the angle measure of the third angle:

Our third angle is and we have a triangle.

A triangle has sides that are in the corresponding ratio of . In this case, the side opposite our angle is , so

We also now know that

Now we know all of our missing side lengths.  The right and left side of the parallelogram will each be . The bottom and top will each be . Let's combine them to find the perimeter:

### Example Question #1 : How To Find The Perimeter Of A Parallelogram

Find the perimeter of the following parallelogram:

Explanation:

The formula for the perimeter of a trapezoid is:

,

where  is the length of the base and  is the length of the edge.

Opposite sides of a parallelogram have the same length. Therefore, both edges are  and both bases are .

Plugging in our values, we get:

### Example Question #2 : How To Find The Perimeter Of A Parallelogram

Find the perimeter of the following parallelogram:

Explanation:

The formula for the perimeter of a parallelogram is:

where  is the length of the longer side and  is the length of the shorter side.

Plugging in our values, we get:

### Example Question #3 : How To Find The Perimeter Of A Parallelogram

Find the perimeter of the following parallelogram:

Explanation:

The formula for the perimeter of a parallelogram is

.

Plugging in our values, we get:

### Example Question #4 : How To Find The Perimeter Of A Parallelogram

ABCD is a parallelogram. BD = 5. The angles of triangle ABD are all equal. What is the perimeter of the parallelogram?

Explanation:

If all of the angles in triangle ABD are equal and line BD divides the parallelogram, then all angles in triangle BDC must be equal as well.

We now have two equilateral triangles, so all sides of the triangles will be equal.

All sides therefore equal 5.

5+5+5+5 = 20

### Example Question #1 : How To Find An Angle In A Parallelogram

In the parallellogram, what is the value of ?

Explanation:

Opposite angles are equal, and adjacent angles must sum to 180.

Therefore, we can set up an equation to solve for z:

(z – 15) + 2z = 180

3z - 15 = 180

3z = 195

z = 65

Now solve for x:

2= x = 130°

### Example Question #1 : Parallelograms

What is the area of a parallelogram with a base of  and a height of ?

Explanation:

To solve this question you must know the formula for the area of a parallelogram.

In this equation, is the length of the base and is the length of the height. We can plug in the side length for both base and height, as given in the question.

### Example Question #2 : Parallelograms

What is the area of a parallelogram with a base of  and a height of ?

Explanation:

To solve this question you must know the formula for the area of a parallelogram.

The formula is

So we can plug in the side length for both base and height to yield

Perform the multiplication to arrive at the answer of .

### Example Question #3 : Parallelograms

Find the area of the following parallelogram:

Cannot be determined from the given information.

Explanation:

The formula for the area of a parallelogram is:

,

where  is the length of the base and  is the length of the height.

In order to the find the height of the parallelogram, use the formula for a  triangle:

, where  is the side opposite the .

The left side of the parallelogram forms the following  triangle:

, where  is the length of the height.

Plugging in our values, we get:

### Example Question #4 : Parallelograms

Find the area of the following parallelogram:

Explanation:

Use the Pythagorean Theorem to determine the length of the diagonal:

The area of the parallelogram is twice the area of the right triangles:

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