Other Polygons - ISEE Upper Level Mathematics Achievement

Card 0 of 19

Question

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

Answer

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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Question

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

Answer

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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Question

What is the sum of all the interior angles of a decagon (a polygon with ten sides)?

Answer

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.

$t=8\cdot 180$

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Question

If each angle in a pentagon is equal to , what is the value of $\frac{1}{2}x$ ?

Answer

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.

Given that a hexagon has 6 angles, the total number of angles will be:

$t=3\cdot 180$

To find the value of each angle, we divide 540 by 5. This results in 108 degrees.

Thus,

$\frac{1}{2}x=54$

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Question

What is the value of an angle (to the nearest degree) in a polygon with sides if all the angles are equal to one another?

Answer

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.

Given that a hexagon has 6 angles, the total number of angles will be:

$t=20\cdot 180$

Given that there are 3,600 degrees total in a polygon with 22 sides, the number of degrees in each angle can be found by dividing 3,600 by 22. To the nearest degree, this results in 164 degrees. Therefore, 164 is the correct answer.

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Question

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

Answer

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.

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

$t=7\cdot 180$

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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Question

Note: Figure NOT drawn to scale.

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

Answer

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

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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Question

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

Answer

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.

$\angle 2$ 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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Question

Note: Figure NOT drawn to scale.

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

Answer

The sum of the degree measures of the angles of Quadrilateral 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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Question

The seven-sided polygon - or heptagon - in the above diagram is regular. What is the measure of $\angle 1$ ?

Answer

In the diagram below, some other angles have been numbered for the sake of convenience.

An interior angle of a regular heptagon has measure

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

This is the measure of $\angle 2$ .

As a result of the Isosceles Triangle Theorem, $\angle 3 \cong \angle 4$ , so

$m \angle 3 = \frac{180^{\circ} - m \angle 2}{2} = \frac{180^{\circ} - 128\frac{4}{7}^{\circ }}{2}= 25\frac{5}{7}^{\circ }$ .

This is also the measure of $\angle 5$ .

By angle addition,

$m\angle 6 = 128\frac{4}{7}^{\circ } - 2 \cdot 25\frac{5}{7}^{\circ } = 77\frac{1}{7}^{\circ }$

Again, as a result of the Isosceles Triangle Theorem, $\angle 1 \cong \angle 7$ , so

$m \angle 1 = \frac{180^{\circ} - m \angle 6}{2} = \frac{180^{\circ} - 77\frac{1}{7}^{\circ }^{\circ }}{2}= 51\frac{3}{7}^{\circ }$

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Question

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

Answer

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

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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Question

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?

Answer

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

Their sum is

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

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

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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Question

Note: Figure NOT drawn to scale.

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

Answer

The sum of the degree measures of the angles of Quadrilateral 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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Question

The measures of the angles of a ten-sided polygon, or decagon, form an arithmetic sequence. The least of the ten degree measures is $100 ^{\circ }$ . What is the greatest of the ten degree measures?

Answer

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

$\left (10-2 \right )180^{\circ} =1,440^{\circ}$ .

In an arithmetic sequence, the terms are separated by a common difference, which we will call . Since the least of the degree measures is $100^{\circ }$ , the measures of the angles are

Their sum is

$100+ \left (100+d \right )+\left (100+2d \right ) ...\left (100+9d \right )=1,440$

$d = 9\frac{7}{9}$

The greatest of the angle measures is

$100+9d = 100 + 9\left ( 9\frac{7}{9}\right )= 100+88 = 188$

However, an angle measure cannot exceed $180^{\circ}$ . The correct choice is that this polygon cannot exist.

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Question

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?

Answer

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

Their sum is

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

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

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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Question

The perimeter of a regular octagon is one mile. Give the sidelength in feet.

Answer

One mile is equal to 5,280 feet, so divide this by 8:

$5,280 \div 8 = 660$ feet

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Question

What is the sidelength, in feet and inches, of a regular octagon with perimeter feet?

Answer

feet is equivalent to $77 \times 12 = 924$ inches. Each side of a regular octagon, an eight-sided figure, measures one-eighth of this:

Each side of an octagon with perimeter feet measures

$924 \div 8 = 115\frac{1}{2}$ inches.

Since

$115\frac{1}{2} \div 12 = 9 \textrm{ R } 7 \frac{1}{2}$ ,

this is equal to $9 \textrm{ ft } 7\frac{ 1}{2} \textrm{ in }$ .

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Question

In a septagon, each of the sides are equal to one another. If the perimeter is feet, what is the length of one of the sides?

Answer

If the perimeter of a septagon, in which each side is equal to the other, is 98, the length of each side will be 7 feet because 98 divided by 7 is 14.

$P=7\cdot s$

$98=7\cdot s$

$\frac{98}{7}=s$

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Question

If the perimeter of a septagon is equal to $5(x^{2}+1)+2x(x-4)$ , what is the length of one side? (All sides of the septagon are equal.)

Answer

If the perimeter of a septagon (in which all sides are equal) is $5(x^{2}+1)+2x(x-4)$ , then the length of one side will be one seventh of this expression.

To find one seventh, the value must first be simplified and then divided by 7.

$5(x^{2}+1)+2x(x-4)$

$5x^{2}+5+2x^{2}-8x$

$7x^{2}-8x+5$

When this is divided by 7, the result is:

$x^{2}-\frac{8x}{7}+\frac{5}{7}$

This value is thus the length of one side of the septagon.

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