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ISEE Upper Level Math : How to find an angle in other polygons

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Example Question #1 : How To Find An Angle In Other Polygons

How many degrees are in an internal angle of a regular heptagon?

Possible Answers:

108

128.6

120

Correct answer:

128.6

Explanation:

The number of degrees in an internal angle of a regular polygon can be solved using the following equation where n equals the number of sides in the polygon:

$\frac{180(n-2)}{n}=\frac{180(7-2)}{7}=\frac{900}{7}=128.6$

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Example Question #1 : How To Find An Angle In Other Polygons

What is the measure of an interior angle of a regular nonagon?

Possible Answers:

128.6

120

140

135

Correct answer:

140

Explanation:

The measure of an interior angle of a regular polygon can be determined using the following equation where n equals the number of sides:

$int=\frac{180(n-2)}{n}=\frac{180(9-2)}{9}=\frac{1260}{9}=140$

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Example Question #1 : Other Polygons

Thingy

Note: Figure NOT drawn to scale.

Refer to the above diagram. Pentagon PENTA is regular. What is the measure of $\angle 1$ ?

Possible Answers:

$136^{\circ }$

$120^{\circ}$

$108^{\circ}$

$128^{\circ}$

$144^{\circ}$

Correct answer:

$144^{\circ}$

Explanation:

The answer can be more clearly seen by extending $\overline{PA}$ to a ray $\overrightarrow{PA}$ :

Thingy

Note that angles have been newly numbered.

$\angle 2$ and $\angle 4$ are exterior angles of a (five-sided) regular pentagon in relation to two parallel lines, so each has a measure of $\left (360 \div 5 \right )^{\circ}= 72^{\circ}$ . $\angle 3$ is a corresponding angle to $\angle 4$ , so its measure is also $72^{\circ}$ .

By angle addition,

$m \angle 1 = m\angle 2 + m\angle 3 = 72^{\circ}+ 72^{\circ} = 144^{\circ}$

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Example Question #17 : Geometry

Thingy

In the above figure, the seven-side polygon, or heptagon, shown is regular. What is the measure of $\angle 1$ ?

Possible Answers:

$102\frac{6}{7}^{\circ}^{\circ}$

The correct answer is not given among the other responses.

$128\frac{4}{7}^{\circ}^{\circ}$

$141\frac{3}{7}^{\circ }$

$154\frac{2}{7}^{\circ}$

Correct answer:

$154\frac{2}{7}^{\circ}$

Explanation:

The answer can be more clearly obtained by extending the top of the two parallel lines as follows:

Note that two angles have been newly labeled.

Thingy

$\angle 2$ is an interior angle of a regular heptagon and therefore has measure

$\frac{\left ( 7-2\right )180^{\circ }}{7}= 128\frac{4}{7}^{\circ }$

By the Isosceles Triangle Theorem, since the two sides of the heptagon that help form the triangle are congruent, so are the two acute angles, and

$m \angle 3 =\frac{1}{2}\left ( 180 -128\frac{4}{7} \right ) ^{\circ }= 25\frac{5}{7}^{\circ }$

$\angle 1$ is supplementary to $\angle 3$ , so

$m \angle 1 = 180^{\circ } - m \angle 3 = 180^{\circ } -25\frac{5}{7}^{\circ }= 154\frac{2}{7}^{\circ}$

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Example Question #11 : Geometry

Thingy

In the above figure, the seven-side polygon, or heptagon, shown is regular. What is the measure of $\angle 1$ ?

Possible Answers:

$141\frac{3}{7}^{\circ }$

The correct answer is not given among the other responses.

$154\frac{2}{7}^{\circ}$

$102\frac{6}{7}^{\circ}^{\circ}$

$128\frac{4}{7}^{\circ}^{\circ}$

Correct answer:

$102\frac{6}{7}^{\circ}^{\circ}$

Explanation:

The answer can be more clearly seen by extending the lower right side of the heptagon to a ray, as shown:

Thingy

Note that angles have been newly numbered.

$\angle 2$ and $\angle 4$ are exterior angles of a (seven-sided) regular heptagon, so each has a measure of $\left (360 \div 7 \right )^{\circ}= 51\frac{3}{7}^{\circ}$ . $\angle 3$ is a corresponding angle to $\angle 4$ in relation to two parallel lines, so its measure is also $51\frac{3}{7}^{\circ}$ .

By angle addition,

$m \angle 1 = m\angle 2 + m\angle 3 = 51\frac{3}{7}^{\circ}^{\circ}+ 51\frac{3}{7}^{\circ}^{\circ} = 102\frac{6}{7}^{\circ}^{\circ}$

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Example Question #1 : Other Polygons

Pentagon

Note: Figure NOT drawn to scale.

In the above figure, Pentagon PENTA is regular. Give the measure of $\angle XZY$ .

Possible Answers:

$59^{\circ}$

$39 ^{\circ}$

$29 ^{\circ}$

$51 ^{\circ}$

$41 ^{\circ}$

Correct answer:

$41 ^{\circ}$

Explanation:

The sum of the degree measures of the angles of Quadrilateral XEYZ is 360, so

$m \angle XZY+ m \angle EXZ + m \angle E + m \angle EYZ = 360^{\circ}$

Each interior angle of a regular pentagon measures

$\frac{(5-2)180^{\circ}}{5} = 108^{\circ}$ ,

which is therefore the measure of $\angle E$ .

It is also given that $m \angle EXZ = 90^{\circ}$ and $m \angle EYZ = 121^{\circ}$ , so substitute and solve:

$m \angle XZY+ m \angle EXZ + m \angle E + m \angle EYZ = 360^{\circ}$

$m \angle XZY+ 90^{\circ} +108^{\circ} +121^{\circ} = 360^{\circ}$

$m \angle XZY+ 319^{\circ} = 360^{\circ}$

$m \angle XZY=41 ^{\circ}$

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Example Question #20 : Geometry

Pentagon

Note: Figure NOT drawn to scale.

In the above figure, Pentagon PENTA is regular. Give the measure of $\angle YZT$ .

Possible Answers:

$85^{\circ}$

The correct answer is not given among the other responses.

$75^{\circ}$

$71^{\circ}$

$61^{\circ}$

Correct answer:

$85^{\circ}$

Explanation:

The sum of the degree measures of the angles of Quadrilateral TNYZ is 360, so

$m \angle YZT + m \angle N+ m \angle T + m \angle ZYN = 360^{\circ}$ .

Each interior angle of a regular pentagon measures

$\frac{(5-2)180^{\circ}}{5} = 108^{\circ}$ ,

which is therefore the measure of both $\angle N$ and $\angle T$ .

$\angle ZYN$ and $\angle ZYE$ form a linear pair, making them supplementary. Since $m \angle ZYE = 121^{\circ}$ ,

$m \angle ZYN = 180^{\circ} - m \angle ZYE = 180^{\circ} -121^{\circ} = 59^{\circ}$ .

Substitute and solve:

$m \angle YZT + 108 ^{\circ} + 108^{\circ} + 59^{\circ} = 360^{\circ}$

$m \angle YZT + 275^{\circ} = 360^{\circ}$

$m \angle YZT = 85^{\circ}$

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Example Question #1 : How To Find An Angle In Other Polygons

The measures of the angles of an octagon form an arithmetic sequence. The greatest of the eight degree measures is $166^{\circ }$ . What is the least of the eight degree measures?

Possible Answers:

$114^{\circ}$

$144^{\circ}$

$104^{\circ}$

$74^{\circ}$

This octagon cannot exist.

Correct answer:

$104^{\circ}$

Explanation:

The total of the degree measures of any eight-sided polygon is

$\left (8-2 \right )180^{\circ} =1,080^{\circ}$ .

In an arithmetic sequence, the terms are separated by a common difference, which we will call . Since the greatest of the degree measures is $166^{\circ }$ , the measures of the angles are

166-7d, 166-6d...166-d, 166

Their sum is

$\left (166-7d \right )+\left ( 166-6d \right ) +...+ \left (166-d \right )+166 = 1,080$

1,328 -28d = 1,080

28d=248

$d= 8\frac{6}{7}$

The least of the angle measures is

$166-7d = 166 - 7 \cdot 8\frac{6}{7} = 166 - 62 = 104$

The correct choice is $104^{\circ}$ .

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Example Question #2 : How To Find An Angle In Other Polygons

What is $\frac{1}{7}$ of the total number of degrees in a 9-sided polygon?

Possible Answers:

360

170

180

Correct answer:

180

Explanation:

The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.

t=(n-2)(180)

Therefore, the equation for the sum of the angles in a 9 sided polygon would be:

t=(9-2)(180)

$t=7\cdot 180$

Therefore, $\frac{1}{7}$ of the total sum of degrees in a 9 sided polygon would be equal to 180 degrees.

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Example Question #11 : Other Polygons

The measures of the angles of a nine-sided polygon, or nonagon, form an arithmetic sequence. The least of the nine degree measures is $127 ^{\circ }$ . What is the greatest of the nine degree measures?

Possible Answers:

$15 2\frac{5}{7}^{\circ}$

$167\frac{6}{11} ^{\circ }$

$147^{\circ}$

$143^{\circ}$

$153^{\circ}$

Correct answer:

$153^{\circ}$

Explanation:

The total of the degree measures of any nine-sided polygon is

$\left (9-2 \right )180^{\circ} = 1,260^{\circ}$ .

In an arithmetic sequence, the terms are separated by a common difference, which we will call . Since the least of the degree measures is $127 ^{\circ }$ , the measures of the angles are

127, 127+d, 127+2d, ...127+8d

Their sum is

$127+\left (127+d \right )+\left ( 127+2d \right )+...+\left (127+8d \right )= 1,260$

1,143+36d = 1,260

36d = 117

$d = 3\frac{1}{4}$

The greatest of the angle measures, in degrees, is

$127+8d = 127+8 \left ( 3\frac{1}{4}\right )= 127 +26 = 153$

$153^{\circ}$ is the correct choice.

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ISEE Upper Level Math : How to find an angle in other polygons

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : How To Find An Angle In Other Polygons

Example Question #1 : How To Find An Angle In Other Polygons

Example Question #1 : Other Polygons

Example Question #17 : Geometry

Example Question #11 : Geometry

Example Question #1 : Other Polygons

Example Question #20 : Geometry

Example Question #1 : How To Find An Angle In Other Polygons

Example Question #2 : How To Find An Angle In Other Polygons

Example Question #11 : Other Polygons

All ISEE Upper Level Math Resources

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