Trapezoids - ACT Math

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Question

Find the measure of angle in the isosceles trapezoid pictured below.

Answer

The sum of the angles in any quadrilateral is 360°, and the properties of an isosceles trapezoid dictate that the sets of angles adjoined by parallel lines (in this case, the bottom set and top set of angles) are equal. Subtracting 2(72°) from 360° gives the sum of the two top angles, and dividing the resulting 216° by 2 yields the measurement of x, which is 108°.

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Question

Given the following isosceles triangle:

In degrees, find the measure of the sum of $\angle x$ and $\angle$ in the figure above.

Answer

All quadrilaterals' interior angles sum to 360°. In isosceles trapezoids, the two top angles are equal to each other.

Similarly, the two bottom angles are equal to each other as well.

Therefore, to find the sum of the two bottom angles, we subtract the measures of the top two angles from 360:

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Question

In the isosceles trapezoid above,

$\overline{AB}\left | \right | \overline{DC}$ . $\angle BDC = 50^{\circ}$ and $\angle BCA = 15^{\circ}$ .

In degrees, what is the measure of $\angle DAC$ ?

Answer

To find the measure of angle DAC, we must know that the interior angles of all triangles sum up to 180 degrees. Also, as this is an isosceles trapezoid, $\angle ADC$ and $\angle BCD$ are equal to each other. The two diagonals within the trapezoid bisect angles $\angle ADC$ and $\angle BCD$ at the same angle.

Thus, $\angle ACD$ must also be equal to 50 degrees.

Thus, $\angle ADC = 50 + 15 = 65$ .

Now that we know two angles out of the three in the triangle on the left, we can subtract them from 180 degrees to find $\angle DAC$ :

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Question

Trapezoid is an isosceles trapezoid with angle $\angle C = 35^{\circ}$ . If $\angle A$ and $\angle B$ are paired, what is the measure of $\angle A$ ?

Answer

As a rule, adjacent (non-paired) angles in a trapezoid are supplementary. Thus, we know that if $\angle C = 35^{\circ}$ , then $\angle B = 145^{\circ}$ . Since we are told that $\angle A$ and $\angle B$ are paired and trapezoid is isosceles, $\angle A$ must also equal $145^{\circ}$ .

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Question

What is the area of this regular trapezoid?

Answer

To solve this question, you must divide the trapezoid into a rectangle and two right triangles. Using the Pythagorean Theorem, you would calculate the height of the triangle which is 4. The dimensions of the rectangle are 5 and 4, hence the area will be 20. The base of the triangle is 3 and the height of the triangle is 4. The area of one triangle is 6. Hence the total area will be 20+6+6=32. If you forget to split the shape into a rectangle and TWO triangles, or if you add the dimensions of the trapezoid, you could arrive at 26 as your answer.

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Question

What is the area of the trapezoid above if a = 2, b = 6, and h = 4?

Answer

Area of a Trapezoid = ½(a+b)h

= ½ (2+6) 4

= ½ (8) * 4

= 4 * 4 = 16

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Question

Find the area of a trapezoid if the height is , and the small and large bases are and , respectively.

Answer

Write the formula to find the area of a trapezoid.

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

Substitute the givens and evaluate the area.

$A= \frac{1}{2} (b1+b2)h=\frac{1}{2} (2x+4x)(2x) = 2x^2+4x^2=6x^2$

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Question

Trapezoid has an area of . If height and , what is the measure of ?

Answer

The formula for the area of a trapezoid is:

$A = \frac{b_1 + b_2}{2}* h$

We have here the height and one of the bases, plus the area, and we are being asked to find the length of base . Plug in known values and solve.

$525 = \frac{37 + b_2}{2}* 21$

$25 = \frac{37 + b_2}{2}$

Thus,

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Question

Find the area of a trapezoid given bases of length 6 and 7 and height of 2.

Answer

To solve, simply use the formula for the area of a trapezoid.

Substitute

into the area formula.

Thus,

$A=\frac{1}{2}(B1+B2)h=\frac{1}{2}(6+7)*2=6+7=13$

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Question

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 ?

Answer

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.

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

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Question

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 $\overline{EF}$ ?

Answer

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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Question

Find the measure of angle in the isosceles trapezoid pictured below.

Answer

The sum of the angles in any quadrilateral is 360°, and the properties of an isosceles trapezoid dictate that the sets of angles adjoined by parallel lines (in this case, the bottom set and top set of angles) are equal. Subtracting 2(72°) from 360° gives the sum of the two top angles, and dividing the resulting 216° by 2 yields the measurement of x, which is 108°.

Compare your answer with the correct one above

Question

Given the following isosceles triangle:

In degrees, find the measure of the sum of $\angle x$ and $\angle$ in the figure above.

Answer

All quadrilaterals' interior angles sum to 360°. In isosceles trapezoids, the two top angles are equal to each other.

Similarly, the two bottom angles are equal to each other as well.

Therefore, to find the sum of the two bottom angles, we subtract the measures of the top two angles from 360:

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Question

In the isosceles trapezoid above,

$\overline{AB}\left | \right | \overline{DC}$ . $\angle BDC = 50^{\circ}$ and $\angle BCA = 15^{\circ}$ .

In degrees, what is the measure of $\angle DAC$ ?

Answer

To find the measure of angle DAC, we must know that the interior angles of all triangles sum up to 180 degrees. Also, as this is an isosceles trapezoid, $\angle ADC$ and $\angle BCD$ are equal to each other. The two diagonals within the trapezoid bisect angles $\angle ADC$ and $\angle BCD$ at the same angle.

Thus, $\angle ACD$ must also be equal to 50 degrees.

Thus, $\angle ADC = 50 + 15 = 65$ .

Now that we know two angles out of the three in the triangle on the left, we can subtract them from 180 degrees to find $\angle DAC$ :

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Question

Trapezoid is an isosceles trapezoid with angle $\angle C = 35^{\circ}$ . If $\angle A$ and $\angle B$ are paired, what is the measure of $\angle A$ ?

Answer

As a rule, adjacent (non-paired) angles in a trapezoid are supplementary. Thus, we know that if $\angle C = 35^{\circ}$ , then $\angle B = 145^{\circ}$ . Since we are told that $\angle A$ and $\angle B$ are paired and trapezoid is isosceles, $\angle A$ must also equal $145^{\circ}$ .

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Question

What is the area of this regular trapezoid?

Answer

To solve this question, you must divide the trapezoid into a rectangle and two right triangles. Using the Pythagorean Theorem, you would calculate the height of the triangle which is 4. The dimensions of the rectangle are 5 and 4, hence the area will be 20. The base of the triangle is 3 and the height of the triangle is 4. The area of one triangle is 6. Hence the total area will be 20+6+6=32. If you forget to split the shape into a rectangle and TWO triangles, or if you add the dimensions of the trapezoid, you could arrive at 26 as your answer.

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Question

What is the area of the trapezoid above if a = 2, b = 6, and h = 4?

Answer

Area of a Trapezoid = ½(a+b)h

= ½ (2+6) 4

= ½ (8) * 4

= 4 * 4 = 16

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Question

Find the area of a trapezoid if the height is , and the small and large bases are and , respectively.

Answer

Write the formula to find the area of a trapezoid.

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

Substitute the givens and evaluate the area.

$A= \frac{1}{2} (b1+b2)h=\frac{1}{2} (2x+4x)(2x) = 2x^2+4x^2=6x^2$

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Question

Trapezoid has an area of . If height and , what is the measure of ?

Answer

The formula for the area of a trapezoid is:

$A = \frac{b_1 + b_2}{2}* h$

We have here the height and one of the bases, plus the area, and we are being asked to find the length of base . Plug in known values and solve.

$525 = \frac{37 + b_2}{2}* 21$

$25 = \frac{37 + b_2}{2}$

Thus,

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Question

Find the area of a trapezoid given bases of length 6 and 7 and height of 2.

Answer

To solve, simply use the formula for the area of a trapezoid.

Substitute

into the area formula.

Thus,

$A=\frac{1}{2}(B1+B2)h=\frac{1}{2}(6+7)*2=6+7=13$

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