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ACT Math : How to find an angle in a kite

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Example Question #241 : Geometry

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:

$257$^{\circ}$$

$103$^{\circ}$$

$128.5$^{\circ}$$

$130.5$^{\circ}$$

$65.5$^{\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 #242 : Geometry

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:

$125$^{\circ}$$

$235$^{\circ}$$

$245$^{\circ}$$

$135$^{\circ}$$

$117.5$^{\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 #243 : 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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Example Question #11 : How To Find An Angle In A Kite

In a particular kite, one angle that lies between congruent sides measures $50^{\circ}$ , and one angle that lies between non-congruent sides measures $25^{\circ}$ . What is the measure of the angle opposite the $50^{\circ}$ angle?

Possible Answers:

$25^{\circ}$

$225^{\circ}$

$260^{\circ}$

$50^{\circ}$

$80^{\circ}$

Correct answer:

$260^{\circ}$

Explanation:

One of the rules governing kites is that the angles which lie between non-congruent sides are congruent to each other. Thus, we know one of the missing angles is also $25^{\circ}$ . Since all angles in a quadrilateral must sum to $360^{\circ}$ , we know that the other missing angle is

$\angle x = 360^{\circ} - (25^{\circ} + 25^{\circ} + 50^{\circ}) = 260^{\circ}$

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ACT Math : How to find an angle in a kite

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #241 : Geometry

Example Question #242 : Geometry

Example Question #243 : Geometry

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

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