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ACT Math : Plane Geometry

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Example Questions

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

Parallelogram_4

In parallelogram ABCD , $\measuredangle B = \left (2x + 15 \right )^{\circ}$ and $\measuredangle C = x^{\circ}$ . Find .

Possible Answers:

115

Correct answer:

Explanation:

In a parallelogram, consecutive angles are supplementary. Thus,

$\measuredangle B + \measuredangle C = 180^{\circ}$

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

3x + 15 = 180

3x = 165

x = 55

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Example Question #1 : Parallelograms

Parallelogram_5

ABCD is a parallelogram. Find .

Possible Answers:

119

Correct answer:

Explanation:

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

1. $\measuredangle B = \measuredangle D$

2. $\measuredangle C + \measuredangle D = 180^{\circ}$

Starting with equation 1.,

2x + 3 = y + x

x + 3 = y

x = y - 3

Now substituting into equation 2.,

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

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

3y - 3 = 180

3y = 183

y = 61

Finally, because opposite angles are congruent, we know that $\measuredangle A = \measuredangle C$ .

z = y

z = 61

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

Parallelogram_7

ABCD is a parallelogram. Find .

Possible Answers:

Correct answer:

Explanation:

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

1. $\measuredangle B =\measuredangle D$

2. $\measuredangle B + \measuredangle C = 180^{\circ}$

3. $\measuredangle A = \measuredangle C$

Starting with equation 1.,

x = 4y - x

2x = 4y

x = 2y

Substituting into equation 2.,

x + y = 180

2y + y = 180

3y = 180

y = 60

Using equation 3.,

z = y

z = 60

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

Parallelogram_2

In parallelogram ABCD , $\measuredangle C = 75^{\circ}$ . What is $\measuredangle D$ ?

Possible Answers:

$105^{\circ}$

$75^{\circ}$

$125^{\circ}$

$90^{\circ}$

$115^{\circ}$

Correct answer:

$105^{\circ}$

Explanation:

In a parellelogram, consecutive angles are supplementary.

$\measuredangle C + \measuredangle D = 180^{\circ}$

$75^{\circ} + \measuredangle D = 180^{\circ}$

$\measuredangle D = 105^{\circ}$

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

Parallelogram_2

In parallelogram ABCD , $\measuredangle D = 125^{\circ}$ . What is $\measuredangle B$ ?

Possible Answers:

$120^{\circ}$

$90^{\circ}$

$125^{\circ}$

$180^{\circ}$

$115^{\circ}$

Correct answer:

$125^{\circ}$

Explanation:

In a parallelogram, opposite angles are congruent.

$\measuredangle B = \measuredangle D$

$\measuredangle B = 125^{\circ}$

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

Parallelogram_8

ABCD is a parallelogram. Find .

Possible Answers:

85.5

Correct answer:

Explanation:

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

$\measuredangle A + \measuredangle D = 180^{\circ}$

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

3z + 9 = 180

3z = 171

z = 57

$\measuredangle A = \measuredangle C$

z = y

57 = y

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

Parallelogram_2

In parallelogram ABCD , $\measuredangle A = 45^{\circ}$ . What is $\measuredangle B?$

Possible Answers:

$180^{\circ}$

$90^{\circ}$

$120^{\circ}$

$135^{\circ}$

$45^{\circ}$

Correct answer:

$135^{\circ}$

Explanation:

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

$\measuredangle A + \measuredangle B = 180^{\circ}$

$45^{\circ} + \measuredangle B = 180^{\circ}$

$\measuredangle B = 135^{\circ}$

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

Parallelogram_2

In parallelogram ABCD , $\measuredangle B = 117^{\circ}$ . What is $\measuredangle D$ ?

Possible Answers:

$63^{\circ}$

$117^{\circ}$

$135^{\circ}$

$180^{\circ}$

$90^{\circ}$

Correct answer:

$117^{\circ}$

Explanation:

In parallelogram ABCD , $\measuredangle B$ and $\measuredangle D$ are opposite angles. In a parallelogram, opposite angles are congruent. This means these two angles are equal.

$\measuredangle D = \measuredangle B$

$\measuredangle D = 117^{\circ}$

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

Parallelogram_2

In parallelogram ABCD , what is the sum of $\measuredangle A$ and $\measuredangle D$ ?

Possible Answers:

$270^{\circ}$

$180^{\circ}$

$90^{\circ}$

$135^{\circ}$

$360^{\circ}$

Correct answer:

$180^{\circ}$

Explanation:

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

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

Parallelogram_6

ABCD is a parallelogram. Find y + z .

Possible Answers:

180

120

140

100

Correct answer:

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. $\measuredangle A + \measuredangle B = 180^{\circ}$

2. $\measuredangle B = \measuredangle D$

3. $\measuredangle C + \measuredangle D = 180^{\circ}$

Start with equation 1.

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

x + 2z = 180

Now simplify equation 2.

x + z = y + 2x

z = y + x

Finally, simplify equation 3.

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

2y + 2x + 20 = 180

2y + 2x = 160

y + x = 80

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

z = y + x = 80

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

x + 2z = 180

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

x + 160 = 180

x = 20

With , we can solve for using equation 3.

y + x = 80

y + 20 = 80

y = 60

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

y + z = 60 + 80 = 140

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ACT Math : Plane Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

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

Example Question #1 : Parallelograms

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

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

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

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

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

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

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

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

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