ACT Math : Plane Geometry

Study concepts, example questions & explanations for ACT Math

varsity tutors app store varsity tutors android store varsity tutors ibooks store

Example Questions

Example Question #1 : Parallelograms

Parallelogram_4

In parallelogram \displaystyle ABCD and . Find \displaystyle x.

Possible Answers:

\displaystyle 45

\displaystyle 115

\displaystyle 55

\displaystyle 25

\displaystyle 65

Correct answer:

\displaystyle 55

Explanation:

In a parallelogram, consecutive angles are supplementary. Thus,

\displaystyle \left ( 2x + 15\right ) + x = 180

\displaystyle 3x + 15 = 180

\displaystyle 3x = 165

\displaystyle x = 55

Example Question #1 : Parallelograms

Parallelogram_5

\displaystyle ABCD is a parallelogram. Find \displaystyle z.

Possible Answers:

\displaystyle 119

\displaystyle 73

\displaystyle 58

\displaystyle 35

\displaystyle 61

Correct answer:

\displaystyle 61

Explanation:

In a parallelogram, consecutive angles are supplementary (i.e. add to \displaystyle 180^{\circ}) and opposite angles are congruent (i.e. equal). Using these properties, we can write a system of equations.

1. 

2. 

 

Starting with equation 1.,

\displaystyle 2x + 3 = y + x

\displaystyle x + 3 = y

\displaystyle x = y - 3

 

Now substituting into equation 2.,

\displaystyle y + \left ( y + x\right ) = 180

\displaystyle 2y + x = 2y + (y - 3) = 180

\displaystyle 3y - 3 = 180

\displaystyle 3y = 183

\displaystyle y = 61

 

Finally, because opposite angles are congruent, we know that .

\displaystyle z = y

\displaystyle z = 61

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

Parallelogram_7

\displaystyle ABCD is a parallelogram. Find \displaystyle z.

Possible Answers:

\displaystyle 45

\displaystyle 60

\displaystyle 80

\displaystyle 30

\displaystyle 90

Correct answer:

\displaystyle 60

Explanation:

In a parallelogram, consecutive angles are supplementary and opposite angles are congruent. Using these properties, we can write a system of equations.

1. 

2. 

3. 

 

Starting with equation 1.,

\displaystyle x = 4y - x

\displaystyle 2x = 4y

\displaystyle x = 2y

 

Substituting into equation 2.,

\displaystyle x + y = 180

\displaystyle 2y + y = 180

\displaystyle 3y = 180

\displaystyle y = 60

 

Using equation 3.,

\displaystyle z = y

\displaystyle z = 60

Example Question #3 : How To Find An Angle In A Parallelogram

Parallelogram_2

In parallelogram \displaystyle ABCD. What is ?

Possible Answers:

\displaystyle 90^{\circ}

\displaystyle 115^{\circ}

\displaystyle 125^{\circ}

\displaystyle 75^{\circ}

\displaystyle 105^{\circ}

Correct answer:

\displaystyle 105^{\circ}

Explanation:

In a parellelogram, consecutive angles are supplementary.

Example Question #2 : How To Find An Angle In A Parallelogram

Parallelogram_2

In parallelogram \displaystyle ABCD. What is ?

Possible Answers:

\displaystyle 120^{\circ}

\displaystyle 90^{\circ}

\displaystyle 125^{\circ}

\displaystyle 180^{\circ}

\displaystyle 115^{\circ}

Correct answer:

\displaystyle 125^{\circ}

Explanation:

In a parallelogram, opposite angles are congruent.

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

Parallelogram_8

\displaystyle ABCD is a parallelogram. Find \displaystyle y.

Possible Answers:

\displaystyle 57

\displaystyle 60

\displaystyle 44

\displaystyle 85.5

\displaystyle 63

Correct answer:

\displaystyle 57

Explanation:

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

\displaystyle z + \left ( 2z + 9\right ) = 180

\displaystyle 3z + 9 = 180

\displaystyle 3z = 171

\displaystyle z = 57

 

\displaystyle z = y

\displaystyle 57 = y

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

Parallelogram_2

In parallelogram \displaystyle ABCD. What is 

Possible Answers:

\displaystyle 180^{\circ}

\displaystyle 90^{\circ}

\displaystyle 120^{\circ}

\displaystyle 135^{\circ}

\displaystyle 45^{\circ}

Correct answer:

\displaystyle 135^{\circ}

Explanation:

In the above parallelogram,  and  are consecutive angles (i.e. next to each other). In a parallelogram, consecutive angles are supplementary, meaning they add to \displaystyle 180^{\circ}.

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

Parallelogram_2

In parallelogram \displaystyle ABCD. What is ?

Possible Answers:

\displaystyle 63^{\circ}

\displaystyle 117^{\circ}

\displaystyle 135^{\circ}

\displaystyle 180^{\circ}

\displaystyle 90^{\circ}

Correct answer:

\displaystyle 117^{\circ}

Explanation:

In parallelogram \displaystyle ABCD and  are opposite angles. In a parallelogram, opposite angles are congruent. This means these two angles are equal.

Example Question #4 : How To Find An Angle In A Parallelogram

Parallelogram_2

In parallelogram \displaystyle ABCD, what is the sum of  and ?

Possible Answers:

\displaystyle 270^{\circ}

\displaystyle 180^{\circ}

\displaystyle 90^{\circ}

\displaystyle 135^{\circ}

\displaystyle 360^{\circ}

Correct answer:

\displaystyle 180^{\circ}

Explanation:

In a parallelogram, consecutive angles are supplementary.   and  are consecutive, so their sum is \displaystyle 180^{\circ}.

Example Question #11 : How To Find An Angle In A Parallelogram

Parallelogram_6

\displaystyle ABCD is a parallelogram. Find \displaystyle y + z.

Possible Answers:

\displaystyle 180

\displaystyle 120

\displaystyle 140

\displaystyle 100

\displaystyle 80

Correct answer:

\displaystyle 140

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. 

 

Start with equation 1.

\displaystyle z + \left (x + z \right ) = 180

\displaystyle x + 2z = 180

 

Now simplify equation 2.

\displaystyle x + z = y + 2x

\displaystyle z = y + x

 

Finally, simplify equation 3.

\displaystyle \left ( y + 20\right ) + \left ( y+ 2x\right ) = 180

\displaystyle 2y + 2x + 20 = 180

\displaystyle 2y + 2x = 160

\displaystyle y + x = 80

 

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

\displaystyle z = y + x = 80

Now that we have \displaystyle z, we can solve for \displaystyle x using equation 1.

\displaystyle x + 2z = 180

\displaystyle x + 2\left ( 80\right ) = 180

\displaystyle x + 160 = 180

\displaystyle x = 20

With \displaystyle x, we can solve for \displaystyle y using equation 3.

\displaystyle y + x = 80

\displaystyle y + 20 = 80

\displaystyle y = 60

Now that we have \displaystyle y and \displaystyle z, we can solve for \displaystyle y + z.

\displaystyle y + z = 60 + 80 = 140

 

Learning Tools by Varsity Tutors