# ACT Math : Parallelograms

## Example Questions

### Example Question #311 : Act Math

is a parallelogram. Find .

Explanation:

In a parallelogram, consecutive angles are supplementary and opposite angles are congruent. Using these properties, we can write a system of equations. Because we have three variables, we will need three equations.

1.

2.

3.

Now simplify equation 2.

Finally, simplify equation 3.

Note that we can plug this simplified equation 3 directly into the simplified equation 2 to solve for .

Now that we have , we can solve for  using equation 1.

With , we can solve for  using equation 3.

Now that we have  and , we can solve for .

### Example Question #321 : Plane Geometry

is a parallelogram. Find .

Explanation:

In a parallelogram, consecutive angles are supplementary and opposite angles are congruent.

### 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 : How To Find The Length Of The Side Of A Parallelogram

In parallelogram  and . Find .

There is insufficient information to solve the problem.

Explanation:

In a parallelogram, opposite sides are congruent. Thus,

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

In parallelogram  and . Find .

There is insufficient information to solve the problem.

Explanation:

In a parallelogram, opposite sides are congruent.

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

Parallelogram  has an area of . If , find .

There is insufficient information to solve the problem.

Explanation:

The area of a parallelogram is given by:

In this problem, the height is given as  and the area is . Both  and  are bases.

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

is a parallelogram. Find .

There is insufficient information to solve the problem.

Explanation:

is the hypotenuse of the right triangle formed when we draw the height of the parallelogram. Because it is a right triangle, we can use SOH CAH TOA to solve for . With respect to , we know the opposite side of the triangle and we are looking for the hypotenuse. Thus, we can use the sine function to solve for .

### Example Question #5 : How To Find The Length Of The Side Of A Parallelogram

Find the length of the base of a parallelogram with a height of  and an area of .

Explanation:

The formula for the area of a parallelogram is:

By plugging in the given values, we get:

### Example Question #6 : How To Find The Length Of The Side Of A Parallelogram

is a parallelogram with an area of . Find .

There is insufficient information to solve the problem.

Explanation:

In order to find , we must first find . The formula for the area of a parallelogram is:

We are given  as the area and  as the base.

Now, we can use trigonometry to solve for . With respect to , we know the opposite side of the right triangle and we are looking for the hypotenuse. Thus, we can use the sine function.

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

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: