ACT Math : How to find an angle in a kite

Study concepts, example questions & explanations for ACT Math

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Example Questions

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

Kite vt act

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

Possible Answers:

\displaystyle 295$^{\circ}$

\displaystyle 300$^{\circ}$

\displaystyle 75$^{\circ}$

\displaystyle 300.5$^{\circ}$

\displaystyle 147.5$^{\circ}$

Correct answer:

\displaystyle 295$^{\circ}$

Explanation:

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

\displaystyle 180 (n-2) degrees, where \displaystyle n 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: \displaystyle 180(4-2) degrees \displaystyle =180(2) degrees \displaystyle =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 \displaystyle 360 degrees and the sum of the non-congruent opposite angles.

The solution is:

\displaystyle 40+25=65

\displaystyle 360-65=295 degrees

Thus, \displaystyle 295 degrees is the sum of the remaining two opposite angles.

Check:

\displaystyle 295+65=360

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

A kite has one set of opposite interior angles where the two angles measure \displaystyle 28$^{\circ}$ and \displaystyle 84$^{\circ}$, respectively. Find the measurement for one of the two remaining interior angles in this kite. 

Possible Answers:

\displaystyle 112$^{\circ}$

\displaystyle 248$^{\circ}$

\displaystyle 124$^{\circ}$

\displaystyle 84^2$^{\circ}$

Not enough information is provided 

Correct answer:

\displaystyle 124$^{\circ}$

Explanation:

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

\displaystyle 180 (n-2) degrees, where \displaystyle n 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: \displaystyle 180(4-2) degrees \displaystyle =180(2) degrees \displaystyle =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 \displaystyle 360 degrees and the non-congruent opposite angles sum by \displaystyle 2:   

\displaystyle 28+84=112

\displaystyle 360-112=248

This means that \displaystyle 248 is the sum of the remaining two angles, which must be opposite congruent angles. Therefore, the measurement for one of the angles is: 

\displaystyle \frac{248}{2}=124 

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

A kite has one set of opposite interior angles where the two angles measure \displaystyle 118$^{\circ}$ and \displaystyle 51$^{\circ}$, respectively. Find the measurement for one of the two remaining interior angles in this kite. 

Possible Answers:

\displaystyle 59.5$^{\circ}$

\displaystyle 95.5$^{\circ}$

\displaystyle 169$^{\circ}$

\displaystyle 95$^{\circ}$

\displaystyle 191$^{\circ}$

Correct answer:

\displaystyle 95.5$^{\circ}$

Explanation:

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

\displaystyle 180 (n-2) degrees, where \displaystyle n 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: \displaystyle 180(4-2) degrees \displaystyle =180(2) degrees \displaystyle =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 \displaystyle 360 degrees and the non-congruent opposite angles sum by \displaystyle 2:   

\displaystyle 118+51=169

\displaystyle 360-169=191

This means that \displaystyle 191 is the sum of the remaining two angles, which must be opposite congruent angles. Therefore, the measurement for one of the angles is: 

\displaystyle \frac{191}{2}=95.5 

Example Question #2 : How To Find An Angle In A Kite

Kite vt act

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

Possible Answers:

\displaystyle 275$^{\circ}$

\displaystyle 85$^{\circ}$

\displaystyle 285$^{\circ}$

\displaystyle 265$^{\circ}$

\displaystyle 132.5$^{\circ}$

Correct answer:

\displaystyle 285$^{\circ}$

Explanation:

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

\displaystyle 180 (n-2) degrees, where \displaystyle n 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: \displaystyle 180(4-2) degrees \displaystyle =180(2) degrees \displaystyle =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 \displaystyle 360 degrees and the sum of the non-congruent opposite angles.

The solution is:

\displaystyle 45+30=75

\displaystyle 360-75=285 degrees

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

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 \displaystyle 91$^{\circ}$ and \displaystyle 99$^{\circ}$ , respectively. Find the measurement for one of the two remaining interior angles in this kite.

Possible Answers:

\displaystyle 170$^{\circ}$

\displaystyle 72$^{\circ}$

\displaystyle 95$^{\circ}$

\displaystyle 85$^{\circ}$

\displaystyle 190$^{\circ}$

Correct answer:

\displaystyle 85$^{\circ}$

Explanation:

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

\displaystyle 180 (n-2) degrees, where \displaystyle n 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: \displaystyle 180(4-2) degrees \displaystyle =180(2) degrees \displaystyle =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 \displaystyle 360 degrees and the non-congruent opposite angles sum by \displaystyle 2:   

\displaystyle 91+99=190

\displaystyle 360-190=170

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

\displaystyle \frac{170}{2}=85 

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 \displaystyle 82$^{\circ}$ and \displaystyle 58$^{\circ}$, respectively. Find the measurement of the sum of the two remaining interior angles in this kite. 

Possible Answers:

\displaystyle 110$^{\circ}$

\displaystyle 240$^{\circ}$

\displaystyle 105$^{\circ}$

\displaystyle 140$^{\circ}$

\displaystyle 220$^{\circ}$

Correct answer:

\displaystyle 220$^{\circ}$

Explanation:

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

\displaystyle 180 (n-2) degrees, where \displaystyle n 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: \displaystyle 180(4-2) degrees \displaystyle =180(2) degrees \displaystyle =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 \displaystyle 360 degrees and the sum of the non-congruent opposite angles.

The solution is:

\displaystyle 82+58=140

\displaystyle 360-140=220 degrees

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

Check:

\displaystyle 110+110+58+82=360

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 \displaystyle 204$^{\circ}$ and \displaystyle 36$^{\circ}$, respectively. Find the measurement for one of the two remaining interior angles in this kite. 

Possible Answers:

\displaystyle 60$^{\circ}$

\displaystyle 120$^{\circ}$

\displaystyle \sqrt{240}$^{\circ}$ 

\displaystyle 140$^{\circ}$

\displaystyle 240$^{\circ}$

Correct answer:

\displaystyle 60$^{\circ}$

Explanation:

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

\displaystyle 180 (n-2) degrees, where \displaystyle n 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: \displaystyle 180(4-2) degrees \displaystyle =180(2) degrees \displaystyle =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 \displaystyle 360 degrees and the non-congruent opposite angles sum by \displaystyle 2:   

\displaystyle 204+36=240

\displaystyle 360-240=120

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

\displaystyle \frac{120}{2}=60 

Example Question #2 : How To Find An Angle In A Kite

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

Possible Answers:

\displaystyle 213$^{\circ}$

\displaystyle 67.5$^{\circ}$

\displaystyle 147$^{\circ}$

\displaystyle 101$^{\circ}$

\displaystyle 62$^{\circ}$

Correct answer:

\displaystyle 147$^{\circ}$

Explanation:

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

\displaystyle 180 (n-2) degrees, where \displaystyle n 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: \displaystyle 180(4-2) degrees \displaystyle =180(2) degrees \displaystyle =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 \displaystyle 360 degrees and the sum of the non-congruent opposite angles.

The solution is:

\displaystyle 112+101=213

\displaystyle 360-213=147 degrees

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

Check:

\displaystyle 213+147=360

Example Question #2 : How To Find An Angle In A Kite

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

Possible Answers:

\displaystyle 78.5$^{\circ}$

\displaystyle 204.5$^{\circ}$

\displaystyle 87.5$^{\circ}$

\displaystyle 203$^{\circ}$

\displaystyle 157$^{\circ}$

Correct answer:

\displaystyle 78.5$^{\circ}$

Explanation:

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

\displaystyle 180 (n-2) degrees, where \displaystyle n 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: \displaystyle 180(4-2) degrees \displaystyle =180(2) degrees \displaystyle =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 \displaystyle 360 degrees and the non-congruent opposite angles sum by \displaystyle 2:   

\displaystyle 16+187=203

\displaystyle 360-203=157

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

\displaystyle \frac{157}{2}=78.5 

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

Kite vt act

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

Possible Answers:

\displaystyle 187.5$^{\circ}$

\displaystyle 87.5$^{\circ}$

\displaystyle 264$^{\circ}$

\displaystyle 106$^{\circ}$

\displaystyle 175$^{\circ}$

Correct answer:

\displaystyle 175$^{\circ}$

Explanation:

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

\displaystyle 180 (n-2) degrees, where \displaystyle n 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: \displaystyle 180(4-2) degrees \displaystyle =180(2) degrees \displaystyle =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 \displaystyle 360 degrees and the sum of the non-congruent opposite angles.

The solution is:

\displaystyle 53+132=185 degrees

\displaystyle 360-185=175 degrees

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

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