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ACT Math : Quadrilaterals

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Example Questions

Example Question #302 : Act Math

Given the height of a trapezoid is and a base length is , what is the length of the other base if the area of the trapezoid is ?

Possible Answers:

Correct answer:

Explanation:

Write the formula used to find the area of a trapezoid.

$A=\frac{1}{2}h(b1+b2)$

Substitute the given information to the formula and solve for the unknown base.

$10=\frac{1}{2}(1)(1+b2)$

20=1+b2

b2=19

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Example Question #1 : How To Find The Length Of The Side Of A Trapezoid

ABCD is an isosceles trapezoid that is bisected by $\overline{EF}$ .

$\overline{AB} \left | \right | \overline{EF} \left | \right | \overline{DC}$ . If $\overline{AE} = 4$ , $\overline{ED} = 6$ , and $\overline{DC} = 18$ , then what is the length of $\overline{EF}$ ?

Possible Answers:

Correct answer:

Explanation:

We know that all three horizontal lines are parallel to one another. By definition, we can set up a ratio between the lengths of the sides provided to us in the question and the lengths of the two parallel lines:

$\frac{AE}{ED} = \frac{EF}{DC}$

Once we substitute the given information, we get

$\frac{4}{6}=\frac{EF}{18}$

We cross multiply to solve for EF

$EF = \frac{(4)(18)}{6}$

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

Parallelogram_2

In parallelogram ABCD , $\overline{AB} = 15$ and the height is . What is $\measuredangle A$ ?

Possible Answers:

$51.3^{\circ}$

$38.7^{\circ}$

$36.9^{\circ}$

$90.0^{\circ}$

$53.1^{\circ}$

Correct answer:

$53.1^{\circ}$

Explanation:

We can start this problem by drawing the height and labeling the lengths with the given values.

Parallelogram_3

When we do this, we can see that we have drawn a triangle inside the paralellogram including $\measuredangle A$ . Because we know the lengths of two sides of this triangle, we can use trigonometry to find $\measuredangle A$ .

With respect to $\measuredangle A$ , we know the values of the opposite and hypotenuse sides of the triangle. Thus, we can use the sine function to solve for $\measuredangle A$ .

$\sin \left (\measuredangle A\right ) = \frac{\textup{opposite}}{\textup{hypotenuse}}$

$\sin \left (\measuredangle A\right ) = \frac{12}{15}= 0.8$

$\measuredangle A = \arcsin\left ( 0.8\right ) = 53.1^{\circ}$

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Example Question #1 : Parallelograms

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 #3 : How To Find An Angle In A Parallelogram

Parallelogram_2

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

Possible Answers:

$90^{\circ}$

$115^{\circ}$

$125^{\circ}$

$75^{\circ}$

$105^{\circ}$

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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ACT Math : Quadrilaterals

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #302 : Act Math

Example Question #1 : How To Find The Length Of The Side Of A Trapezoid

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

Example Question #1 : Parallelograms

Example Question #1 : Parallelograms

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

Example Question #3 : 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

All ACT Math Resources

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