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ACT Math : Plane Geometry

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Example Questions

Example Question #1 : Quadrilaterals

Kite vt act

Using the kite shown above, find the sum of the two remaining congruent interior angles.

Possible Answers:

$132.5$^{\circ}$$

$285$^{\circ}$$

$265$^{\circ}$$

$85$^{\circ}$$

$275$^{\circ}$$

Correct answer:

$285$^{\circ}$$

Explanation:

The sum of the interior angles of any polygon can be found by applying the formula:

180 (n-2) degrees, where is the number of sides in the polygon.

A kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: 180(4-2) degrees =180(2) degrees =360 degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.

To find the sum of the remaining two angles, determine the difference between 360 degrees and the sum of the non-congruent opposite angles.

The solution is:

45+30=75

360-75=285 degrees

Thus, 285 degrees is the sum of the remaining two opposite angles.

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

A kite has one set of opposite interior angles where the two angles measure $91$^{\circ}$$ and $99$^{\circ}$$ , respectively. Find the measurement for one of the two remaining interior angles in this kite.

Possible Answers:

$95$^{\circ}$$

$72$^{\circ}$$

$85$^{\circ}$$

$170$^{\circ}$$

$190$^{\circ}$$

Correct answer:

$85$^{\circ}$$

Explanation:

The sum of the interior angles of any polygon can be found by applying the formula:

180 (n-2) degrees, where is the number of sides in the polygon.

By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: 180(4-2) degrees =180(2) degrees =360 degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.

The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between 360 degrees and the non-congruent opposite angles sum by :

91+99=190

360-190=170

This means that 170 is the sum of the remaining two angles, which must be opposite congruent angles. Therefore, the measurement for one of the angles is:

$\frac{170}{2}=85$

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Example Question #2 : Quadrilaterals

A kite has one set of opposite interior angles where the two angles measure $82$^{\circ}$$ and $58$^{\circ}$$ , respectively. Find the measurement of the sum of the two remaining interior angles in this kite.

Possible Answers:

$140$^{\circ}$$

$240$^{\circ}$$

$105$^{\circ}$$

$220$^{\circ}$$

$110$^{\circ}$$

Correct answer:

$220$^{\circ}$$

Explanation:

The sum of the interior angles of any polygon can be found by applying the formula:

180 (n-2) degrees, where is the number of sides in the polygon.

By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: 180(4-2) degrees =180(2) degrees =360 degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.

To find the sum of the remaining two angles, determine the difference between 360 degrees and the sum of the non-congruent opposite angles.

The solution is:

82+58=140

360-140=220 degrees

This means that 220 degrees is the sum of the remaining two opposite angles and that each have an individual measurement of 110 degrees.

Check:

110+110+58+82=360

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

A kite has one set of opposite interior angles where the two angles measure $204$^{\circ}$$ and $36$^{\circ}$$ , respectively. Find the measurement for one of the two remaining interior angles in this kite.

Possible Answers:

$140$^{\circ}$$

$60$^{\circ}$$

$240$^{\circ}$$

$\sqrt{240}$^{\circ}$$

$120$^{\circ}$$

Correct answer:

$60$^{\circ}$$

Explanation:

The sum of the interior angles of any polygon can be found by applying the formula:

180 (n-2) degrees, where is the number of sides in the polygon.

A kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: 180(4-2) degrees =180(2) degrees =360 degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.

The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between 360 degrees and the non-congruent opposite angles sum by :

204+36=240

360-240=120

This means that 120 is the sum of the remaining two angles, which must be opposite congruent angles. Therefore, the measurement for one of the angles is:

$\frac{120}{2}=60$

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Example Question #5 : How To Find An Angle In A Kite

A kite has one set of opposite interior angles where the two angles measure $112$^{\circ}$$ and $101$^{\circ}$$ , respectively. Find the measurement of the sum of the two remaining interior angles.

Possible Answers:

$147$^{\circ}$$

$213$^{\circ}$$

$67.5$^{\circ}$$

$62$^{\circ}$$

$101$^{\circ}$$

Correct answer:

$147$^{\circ}$$

Explanation:

The sum of the interior angles of any polygon can be found by applying the formula:

180 (n-2) degrees, where is the number of sides in the polygon.

By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: 180(4-2) degrees =180(2) degrees =360 degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.

To find the sum of the remaining two angles, determine the difference between 360 degrees and the sum of the non-congruent opposite angles.

The solution is:

112+101=213

360-213=147 degrees

This means that 147 degrees is the sum of the remaining two opposite angles.

Check:

213+147=360

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

A kite has one set of opposite interior angles where the two angles measure $16$^{\circ}$$ and $187$^{\circ}$$ , respectively. Find the measurement for one of the two remaining interior angles in this kite.

Possible Answers:

$87.5$^{\circ}$$

$78.5$^{\circ}$$

$203$^{\circ}$$

$157$^{\circ}$$

$204.5$^{\circ}$$

Correct answer:

$78.5$^{\circ}$$

Explanation:

The sum of the interior angles of any polygon can be found by applying the formula:

180 (n-2) degrees, where is the number of sides in the polygon.

By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: 180(4-2) degrees =180(2) degrees =360 degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.

The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between 360 degrees and the non-congruent opposite angles sum by :

16+187=203

360-203=157

This means that 157 is the sum of the remaining two angles, which must be opposite congruent angles. Therefore, the measurement for one of the angles is:

$\frac{157}{2}=78.5$

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Example Question #3 : Quadrilaterals

Kite vt act

Using the kite shown above, find the sum of the two remaining congruent interior angles.

Possible Answers:

$187.5$^{\circ}$$

$264$^{\circ}$$

$106$^{\circ}$$

$87.5$^{\circ}$$

$175$^{\circ}$$

Correct answer:

$175$^{\circ}$$

Explanation:

The sum of the interior angles of any polygon can be found by applying the formula:

180 (n-2) degrees, where is the number of sides in the polygon.

A kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: 180(4-2) degrees =180(2) degrees =360 degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.

To find the sum of the remaining two angles, determine the difference between 360 degrees and the sum of the non-congruent opposite angles.

The solution is:

53+132=185 degrees

360-185=175 degrees

Thus, 175 degrees is the sum of the remaining two opposite angles.

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Example Question #11 : Kites

A kite has one set of opposite interior angles where the two angles measure $44$^{\circ}$$ and $59$^{\circ}$$ , respectively. Find the measurement for one of the two remaining interior angles in this kite.

Possible Answers:

$130.5$^{\circ}$$

$103$^{\circ}$$

$65.5$^{\circ}$$

$257$^{\circ}$$

$128.5$^{\circ}$$

Correct answer:

$128.5$^{\circ}$$

Explanation:

The sum of the interior angles of any polygon can be found by applying the formula:

180 (n-2) degrees, where is the number of sides in the polygon.

By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: 180(4-2) degrees =180(2) degrees =360 degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.

The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between 360 degrees and the non-congruent opposite angles sum by :

44+59=103

360-103=257

This means that 257 is the sum of the remaining two angles, which must be opposite congruent angles. Therefore, the measurement for one of the angles is:

$\frac{257}{2}=128.5$

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Example Question #12 : Kites

A kite has one set of opposite interior angles where the two angles measure $58$^{\circ}$$ and $67$^{\circ}$$ , respectively. Find the measurement of the sum of the two remaining interior angles.

Possible Answers:

$245$^{\circ}$$

$125$^{\circ}$$

$235$^{\circ}$$

$117.5$^{\circ}$$

$135$^{\circ}$$

Correct answer:

$235$^{\circ}$$

Explanation:

The sum of the interior angles of any polygon can be found by applying the formula:

180 (n-2) degrees, where is the number of sides in the polygon.

By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: 180(4-2) degrees =180(2) degrees =360 degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.

To find the sum of the remaining two angles, determine the difference between 360 degrees and the sum of the non-congruent opposite angles.

The solution is:

58+67=125

$360-125=235$^{\circ}$$

This means that $235$^{\circ}$$ is the sum of the remaining two opposite angles.

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Example Question #241 : Geometry

If the diagonals of the quadrilateral above were drawn in the figure, they would form four 90 degree angles at the center. In degrees, what is the value of ?

Possible Answers:

Correct answer:

Explanation:

A quadrilateral is considered a kite when one of the following is true:

(1) it has two disjoint pairs of sides are equal in length or

(2) one diagonal is the perpendicular bisector of the other diagonal. Given the information in the question, we know (2) is definitely true.

To find we must first find the values of all of the angles.

The sum of angles within any quadrilateral is 360 degrees.

Therefore $\angle D=\angle B$ .

To find $\angle A$ :

360-115-115=130

130=2(x+4)=2x+8

$x=\frac{130-8}{2}=\frac{122}{2}=61$

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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 : Quadrilaterals

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

Example Question #2 : Quadrilaterals

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

Example Question #5 : How To Find An Angle In A Kite

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

Example Question #3 : Quadrilaterals

Example Question #11 : Kites

Example Question #12 : Kites

Example Question #241 : Geometry

All ACT Math Resources

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