GMAT Math : Calculating an angle in a polygon

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Example Question #1 : Calculating An Angle In A Polygon

You are given Pentagon ABCDE such that:

$m \angle A = \frac{7}{5} m \angle B$

and

$\angle B \cong \angle C \cong \angle D \cong \angle E$

Calculate $m \angle A$

Possible Answers:

$140^{\circ }$

$151.2^{\circ }$

$108^{\circ }$

$100^{\circ }$

This pentagon cannot exist

Correct answer:

$140^{\circ }$

Explanation:

Let be the common measure of $\angle B$ , $\angle C$ , $\angle D$ , and $\angle E$

Then

$m \angle A = \frac{7}{5} x$

The sum of the measures of the angles of a pentagon is $180 (5 - 2) = 540^{\circ }$ degrees; this translates to the equation

$\frac{7}{5} x + x + x + x + x = 540$

$\frac{27}{5} x = 540$

$x = 540 \cdot \frac{5}{27} = 100$

$m \angle A = \frac{7}{5} x = \frac{7}{5} \cdot 100$

$m \angle A = 140$

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Example Question #2 : Calculating An Angle In A Polygon

Polygons_1

The above diagram shows a regular pentagon and a regular hexagon sharing a side. Give $m\angle AVB$ .

Possible Answers:

$120^{\circ }$

$144^{\circ }$

$132^{\circ }$

$150^{\circ }$

$168^{\circ }$

Correct answer:

$132^{\circ }$

Explanation:

This can more easily be explained if the shared side is extended in one direction, and the new angles labeled.

Polygons_2

$\angle1$ and $\angle 2$ are exterior angles of the regular polygons. Also, the measures of the exterior angles of any polygon, one at each vertex, total $360^{\circ }$ . Therefore,

$m \angle1 = \frac{360}{5} = 72^{\circ }$

$m \angle2 = \frac{360}{6} = 60^{\circ }$

Add the measures of the angles to get $m\angle AVB$ :

$m\angle AVB = m\angle 1 + m\angle 2 = 72 + 60 = 132^{\circ }$

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Example Question #3 : Calculating An Angle In A Polygon

Which of the following cannot be the measure of an exterior angle of a regular polygon?

Possible Answers:

$9^{\circ }$

$6^{\circ }$

$16^{\circ }$

$15^{\circ }$

Each of the given choices can be the measure of an exterior angle of a regular polygon.

Correct answer:

$16^{\circ }$

Explanation:

The sum of the measures of the exterior angles of any polygon, one per vertex, is $360^{\circ }$ . In a regular polygon of sides , then all of these exterior angles are congruent, each measuring $\frac{360^{\circ }}{N}$ .

If is the measure of one of these angles, then $x = \frac{360^{\circ }}{N}$ , or, equivalently, $N = \frac{360^{\circ }}{x}$ . Therefore, for to be a possible measure of an exterior angle, it must divide evenly into 360. We divide each in turn:

$360 \div 6 = 60$

$360 \div 9 = 40$

$360 \div 15 = 24$

$360 \div 16 = 22.5$

Since 16 is the only one of the choices that does not divide evenly into 360, it cannot be the measure of an exterior angle of a regular polygon.

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Example Question #4 : Calculating An Angle In A Polygon

Pentagon_and_square

Note: Figure NOT drawn to scale

The figure above shows a square inside a regular pentagon. Give $m\angle 1$ .

Possible Answers:

$30^{\circ }$

$22^{\circ }$

$15^{\circ }$

$18^{\circ }$

$24^{\circ }$

Correct answer:

$18^{\circ }$

Explanation:

Each angle of a square measures $90^{\circ }$ ; each angle of a regular pentagon measures $180 \cdot3 \div 5 = 108^{\circ }$ . To get $m\angle 1$ , subtract:

$108 - 90 = 18^{\circ }$ .

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Example Question #1 : Calculating An Angle In A Polygon

Hexagon

Note: Figure NOT drawn to scale.

Given:

$\angle A \cong \angle C \cong \angle E$

$\angle B \cong \angle D \cong \angle F$

$m \angle A + 6^{\circ } = m \angle B$

Evaluate $m \angle A$ .

Possible Answers:

$111 ^{ \circ }$

$129 ^{ \circ }$

$114 ^{\circ }$

$117 ^{ \circ }$

$123 ^{ \circ }$

Correct answer:

$117 ^{ \circ }$

Explanation:

Call the measure of $\angle A$

$m \angle A = m \angle C = m \angle E = x$

$m \angle A + 6^{\circ } = m \angle B$ , and $\angle B \cong \angle D \cong \angle F$

$m \angle B = m \angle D = m \angle F = x + 6$

The sum of the measures of the angles of a hexagon is $180 (6 - 2) = 720 ^{ \circ }$ , so

$m \angle A + m \angle B + m \angle C + m \angle D + m \angle E + m \angle F = 720$

x+ (x+6) +x + (x+6) + x + (x+6) = 720

6x + 18 = 720

$\ 6x = 702$

$\ x = 702 \div 6 = 117$ , which is the measure of $\angle A$ .

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Example Question #6 : Calculating An Angle In A Polygon

Which of the following figures would have exterior angles none of whose degree measures is an integer?

Possible Answers:

A regular polygon with ninety sides.

A regular polygon with forty-five sides.

A regular polygon with twenty-four sides.

A regular polygon with thirty sides.

A regular polygon with eighty sides.

Correct answer:

A regular polygon with eighty sides.

Explanation:

The sum of the degree measures of any polygon is $360^{\circ }$ . A regular polygon with sides has exterior angles of degree measure $\left (\frac{360}{N} \right )^{\circ }$ . For this to be an integer, 360 must be divisible by .

We can test each of our choices to see which one fails this test.

$360 \div 24 = 15$

$360 \div 30 = 12$

$360 \div 45 = 8$

$360 \div 80 = 4.5$

$360 \div 90 = 4$

Only the eighty-sided regular polygon fails this test, making this the correct choice.

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Example Question #1 : Calculating An Angle In A Polygon

Thingy

The above diagram shows a regular pentagon and a regular hexagon sharing a side. What is the measure of $\angle ABC$ ?

Possible Answers:

$18 ^{\circ }$

$20 ^{\circ }$

$10 ^{\circ }$

$15 ^{\circ }$

$12 ^{\circ }$

Correct answer:

$12 ^{\circ }$

Explanation:

The measure of each interior angle of a regular pentagon is

$\frac{180 (5-2)}{5} = \frac{180 (3)}{5} = 108^{\circ }$

The measure of each interior angle of a regular hexagon is

$\frac{180 (6-2)}{6} = \frac{180 (4)}{6} = 120^{\circ }$

The measure of $\angle ABC$ is the difference of the two, or $12 ^{\circ }$ .

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Example Question #1 : Calculating An Angle In A Polygon

What is the arithmetic mean of the measures of the angles of a nonagon (a nine-sided polygon)?

Possible Answers:

$135^{\circ }$

The question cannot be answered without knowing the measures of the individual angles.

$108^{\circ }$

$120^{\circ }$

$140^{\circ }$

Correct answer:

$140^{\circ }$

Explanation:

The sum of the measures of the nine angles of any nonagon is calculated as follows:

$180 \cdot \left ( 9-2\right ) =180 \cdot 7= 1,260^{\circ }$

Divide this number by nine to get the arithmetic mean of the measures:

$1,260\div 9 = 140 ^{\circ }$

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Example Question #1 : Calculating An Angle In A Polygon

You are given a quadrilateral and a pentagon. What is the mean of the measures of the interior angles of the two polygons?

Possible Answers:

$96^{\circ }$

$100^{\circ }$

$102^{\circ }$

Insufficient information is given to answer the question.

$99^{\circ }$

Correct answer:

$100^{\circ }$

Explanation:

The mean of the measures of the four angles of the quadrilateral and the five angles of of the pentagon is their sum divided by 9.

The sum of the measures of the interior angles of any quadrilateral is $180 (4-2) = 360^{\circ }$ . The sum of the measures of the interior angles of any pentagon is $180 (5-2) = 540^{\circ }$ .

The sum of the measures of the interior angles of both polygons is therefore $360 + 540 = 900^{\circ }$ . Divide by 9:

$900 \div 9 = 100^{\circ }$

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Example Question #1 : Calculating An Angle In A Polygon

What is the median of the measures of the angles of a nonagon (a nine-sided polygon)?

Possible Answers:

$135^{\circ }$

$120^{\circ }$

$140^{\circ }$

$108^{\circ }$

The question cannot be answered without knowing the measures of the individual angles.

Correct answer:

The question cannot be answered without knowing the measures of the individual angles.

Explanation:

The sum of the measures of the nine angles of any nonagon is calculated as follows:

$180 \cdot \left ( 9-2\right ) =180 \cdot 7= 1,260^{\circ }$

The median of an odd quantity of numbers is the number that falls in the center position when they are arranged in ascending order; for nine numbers, it will be the fifth-highest number. We now need to show that we need to know the actual numbers in order to find the median.

Case 1: Each angle measures $140 ^{\circ }$ .

The set is $\left \{ 140,140,140,140,140,140,140,140,140\right \}$ and the median is 140.

Case 2: Eight of the angles measure $139 ^{\circ }$ and one of them measures $148 ^{\circ }$ .

The set is $\left \{}139, 139, 139, 139, 139, 139, 139, 139, 148 \right \}$ and the median is 139.

In both cases, the sum of the angle measures is 1,260, but the medians differ between the two.

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GMAT Math : Calculating an angle in a polygon

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Example Questions

Example Question #1 : Calculating An Angle In A Polygon

Example Question #2 : Calculating An Angle In A Polygon

Example Question #3 : Calculating An Angle In A Polygon

Example Question #4 : Calculating An Angle In A Polygon

Example Question #1 : Calculating An Angle In A Polygon

Example Question #6 : Calculating An Angle In A Polygon

Example Question #1 : Calculating An Angle In A Polygon

Example Question #1 : Calculating An Angle In A Polygon

Example Question #1 : Calculating An Angle In A Polygon

Example Question #1 : Calculating An Angle In A Polygon

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