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ACT Math : Plane Geometry

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Example Questions

Example Question #2 : Rhombuses

A rhombus has two interior angles each with a measurement of $75$^\circ$$ . Find the sum of the remaining two angles.

Possible Answers:

$200$^\circ$$

$110$^\circ$$

$105$^\circ$$

$210$^\circ$$

$90$^\circ$$

Correct answer:

$210$^\circ$$

Explanation:

To solve this problem, consider that the sum total of the four interior angles in any rhombus must equal $360$^\circ$$ . Furthermore, a rhombus must have two sets of equivalent opposite interior angles, and a rhombus must have two sets of adjacent interior angles. The adjacent interior angles must be supplementary—meaning they have a sum total of $180$^\circ$$ .

Since this problem provides the measurement for two of the interior angles, find the sum of those two angles. Then subtract that sum from $360$^\circ$$ to find the sum of the two remaining interior angles.

The solution is:

75+75=150

$360-150=210$^\circ$$

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Example Question #3 : Rhombuses

A rhombus has two interior angles each with a measurement of $129$^\circ$$ . Find the sum of the remaining two angles.

Possible Answers:

$204$^\circ$$

$51$^\circ$$

$78$^\circ$$

$156$^\circ$$

$102$^\circ$$

Correct answer:

$102$^\circ$$

Explanation:

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Example Question #6 : How To Find An Angle In A Rhombus

A rhombus has an interior angle with a measurement of $140$^\circ$$ . Find the measurement for one of the angles that are adjacent to the angle that has a measurement of $140$^\circ$$ .

Possible Answers:

$20$^\circ$$

$105$^\circ$$

$110$^\circ$$

$40$^\circ$$

$80$^\circ$$

Correct answer:

$40$^\circ$$

Explanation:

A rhombus must have two sets of equivalent opposite interior angles, and a rhombus must have two sets of adjacent interior angles. The adjacent interior angles must be supplementary—meaning they have a sum total of $180$^\circ$$ .

Therefore, if a rhombus has an interior angle that has a measurement of $140$^\circ$$ , then the adjacent angle must equal:

$180-140=40$^\circ$$

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Example Question #4 : Rhombuses

Act rhombus vt

Using the rhombus shown above, find the measurement for angle .

Possible Answers:

$96$^\circ$$

$94$^\circ$$

$90$^\circ$$

$84$^\circ$$

$102$^\circ$$

Correct answer:

$96$^\circ$$

Explanation:

A rhombus must have two sets of equivalent opposite interior angles, and a rhombus must have two sets of adjacent interior angles. The adjacent interior angles must be supplementary—meaning they have a sum total of $180$^\circ$$ .

Since this problem involves supplementary angles, the solution is:

$180-84=96$^\circ$$

Check answer by:

$96+84=180$^\circ$$

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Example Question #11 : Rhombuses

A rhombus has two interior angles each with a measurement of $46$^\circ$$ . Find the measurement for one of the two remaining angles.

Possible Answers:

$64$^\circ$$

$94$^\circ$$

$88$^\circ$$

$124$^\circ$$

$134$^\circ$$

Correct answer:

$134$^\circ$$

Explanation:

First, consider that the sum total of the four interior angles in any rhombus must equal $360$^\circ$$ . Secondly, a rhombus must have two sets of equivalent opposite interior angles, and a rhombus must have two sets of adjacent interior angles. The adjacent interior angles must be supplementary—meaning they have a sum total of $180$^\circ$$ .

One way to approach this problem is to realize that each of the remaining two angles must have the same measurement, and that each will be supplementary angles with $46$^\circ$$ . Therefore, find the difference between $180$^\circ$$ and $46$^\circ$$ to find the solution.

$180-46=134$^\circ$$

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Example Question #201 : Quadrilaterals

A rhombus has two interior angles each with a measurement of $159$^\circ$$ . Find the measurement for one of the two remaining angles.

Possible Answers:

$18$^\circ$$

$19$^\circ$$

$40$^\circ$$

$21$^\circ$$

$42$^\circ$$

Correct answer:

$21$^\circ$$

Explanation:

A rhombus must have two sets of equivalent opposite interior angles, and a rhombus must have two sets of adjacent interior angles. The adjacent interior angles must be supplementary—meaning they have a sum total of $180$^\circ$$ .

One way to approach this problem is to realize that each of the remaining two angles must have the same measurement, and that each will be supplementary angles with $159$^\circ$$ . Find the difference between $180$^\circ$$ and $159$^\circ$$ to find the solution.

$180-159=21$^\circ$$

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Example Question #13 : How To Find An Angle In A Rhombus

A rhombus has two interior angles each with a measurement of $121$^\circ$$ . Find the sum of the remaining two angles.

Possible Answers:

$180$^\circ$$

$59$^\circ$$

$118$^\circ$$

$62$^\circ$$

$90$^\circ$$

Correct answer:

$118$^\circ$$

Explanation:

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Example Question #51 : Rhombuses

If the area of a rhombus is $60\:cm^2$ , and the length of one of its diagonals is $2\:cm$ , what must be the length of the other diagonal?

Possible Answers:

$30\:cm$

$60\:cm$

$90\:cm$

$120\:cm$

$45\:cm$

Correct answer:

$60\:cm$

Explanation:

Write the formula for the area of a rhombus.

$A=\frac{d_1\cdot d_2}{2}$

Plug in the given area and diagonal length. Solve for the other diagonal.

$60\:cm^2=\frac{2\:cm\cdot d_2}{2}$

$2(60\:cm^2)=2(\frac{2\:cm\cdot d_2}{2})$

$120\:cm^2=2\:cm\cdot d_2$

$\frac{120\:cm}{2\:cm}=\frac{2\:cm\cdot d_2}{2\:cm}$

$d_2 =60\:cm$

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Example Question #11 : Rhombuses

A rhombus contains diagonals with the length $12 \:cm$ and $6 \:cm$ . Find the area of the rhombus.

Possible Answers:

$72\:cm^2$

$54\:cm^2$

$96\:cm^2$

$88\:cm^2$

$36\:cm^2$

Correct answer:

$36\:cm^2$

Explanation:

The equation for the area of a rhombus is given by:

$A= \frac{p \cdot q}{2}$

where and are the two diagonal lengths.

This problem very quickly becomes one of the "plug and chug" type, where the given values just need to be substituted into the equation and the equation then solved. By plugging in the values given, we get:

$A = \frac{12\:cm \cdot 6\:cm}{2}$

$A = \frac{72\:cm^2}2{}$

$A= 36\:cm^2$

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Example Question #1 : How To Find The Area Of A Rhombus

Find the area of a rhombus if the diagonals lengths are $20\:cm$ and $40\:cm$ .

Possible Answers:

$900\:cm^2$

$600\:cm^2$

$200\:cm^2$

$800\:cm^2$

$400\:cm^2$

Correct answer:

$400\:cm^2$

Explanation:

Write the formula for the area of a rhombus:

$A=\frac{d_1 \cdot d_2}{2}$

Substitute the given lengths of the diagonals and solve:

$A=\frac{d1 \cdot d2}{2} = \frac{20\:cm \cdot 40\:cm}{2} =\frac{800\:cm^2}{2} = 400\:cm^2$

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ACT Math : Plane Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #2 : Rhombuses

Example Question #3 : Rhombuses

Example Question #6 : How To Find An Angle In A Rhombus

Example Question #4 : Rhombuses

Example Question #11 : Rhombuses

Example Question #201 : Quadrilaterals

Example Question #13 : How To Find An Angle In A Rhombus

Example Question #51 : Rhombuses

Example Question #11 : Rhombuses

Example Question #1 : How To Find The Area Of A Rhombus

All ACT Math Resources

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