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ISEE Upper Level Math : Geometry

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Example Questions

Example Question #3 : 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:

This octagon cannot exist.

$74^{\circ}$

$144^{\circ}$

$114^{\circ}$

$104^{\circ}$

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 #4 : 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:

170

360

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 #21 : Geometry

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:

$147^{\circ}$

$143^{\circ}$

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

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

$153^{\circ}$

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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Example Question #11 : How To Find An Angle In Other Polygons

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?

Possible Answers:

This polygon cannot exist.

$160^{\circ}$

$170^{\circ}$

$116^{\circ}$

$140^{\circ}$

Correct answer:

This polygon cannot exist.

Explanation:

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

100, 100+d, 100+2d, ...100+9d

Their sum is

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

1,000+ 45d =1,440

45d = 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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Example Question #21 : Geometry

Heptagon

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

Possible Answers:

$77\frac{1}{7}^{\circ }$

$70^{\circ}$

$38\frac{4}{7}^{\circ }$

$51\frac{3}{7}^{\circ }$

$60 ^{\circ}$

Correct answer:

$51\frac{3}{7}^{\circ }$

Explanation:

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

Heptagon

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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Example Question #11 : How To Find An Angle In Other Polygons

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

Possible Answers:

1,300

1,250

1,440

1,800

Correct answer:

1,440

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)

t=(10-2)(180)

$t=8\cdot 180$

t=1,440

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Example Question #12 : Other Polygons

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

Possible Answers:

540

108

Correct answer:

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=(5-2)(180)

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

t=(5-2)(180)

$t=3\cdot 180$

t=540

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

Thus,

x=108

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

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Example Question #25 : Plane Geometry

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?

Possible Answers:

$164^\circ$

$180^\circ$

$163^\circ$

$3,600^\circ$

Correct answer:

$164^\circ$

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)

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

t=(22-2)(180)

$t=20\cdot 180$

t=3,600

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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Example Question #1 : How To Find The Length Of A Side In Other Polygons

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

Possible Answers:

$1,320 \textrm{ ft}$

$990 \textrm{ ft}$

$660 \textrm{ ft}$

$330 \textrm{ ft}$

$880 \textrm{ ft}$

Correct answer:

$660 \textrm{ ft}$

Explanation:

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

$5,280 \div 8 = 660$ feet

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Example Question #2 : How To Find The Length Of A Side In Other Polygons

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

Possible Answers:

$7\textup{ ft } 11 \textrm{ in}$

$7\textrm{ ft } 8 \frac{2}{5} \textrm{ in }$

$9 \textrm{ ft } 7\frac{ 1}{2} \textrm{ in }$

$11\textrm{ ft } 8 \textrm{ in }$

$12 \textrm{ ft } 10 \textrm{ in }$

Correct answer:

$9 \textrm{ ft } 7\frac{ 1}{2} \textrm{ in }$

Explanation:

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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ISEE Upper Level Math : Geometry

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #3 : How To Find An Angle In Other Polygons

Example Question #4 : How To Find An Angle In Other Polygons

Example Question #21 : Geometry

Example Question #11 : How To Find An Angle In Other Polygons

Example Question #21 : Geometry

Example Question #11 : How To Find An Angle In Other Polygons

Example Question #12 : Other Polygons

Example Question #25 : Plane Geometry

Example Question #1 : How To Find The Length Of A Side In Other Polygons

Example Question #2 : How To Find The Length Of A Side In Other Polygons

All ISEE Upper Level Math Resources

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