GMAT Math : Polygons

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Example Questions

Example Question #2 : Dsq: Calculating An Angle In A Polygon

What is the sum of six numbers, a,b,c,d,e,f ?

Statement 1: a,b,c,d,e,f are the measures of the interior angles of a hexagon.

Statement 2: The median of the data set $\left \{}a,b,c,d,e,f \right \}$ is 120.

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

The sum of the measures of the six interior angles of any hexagon is 180 (6-2) = 720 , so if Statement 1 is true, then the sum of the six numbers is 720, regardless of the individual values.

Statement 2 is not enough, however, for us to deduce this sum; this only tells us the mean of the third- and fourth-highest values.

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Example Question #3 : Dsq: Calculating An Angle In A Polygon

What is the sum of five numbers, a,b,c,d,e ?

Statement 1: a,b,c,d,e are the measures of the interior angles of a pentagon.

Statement 2: The mean of the data set $\left \{}a,b,c,d,e\right \}$ is 108.

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The sum of the measures of the five interior angles of any pentagon is 180 (5-2) = 540 , so if Statement 1 is true, then the sum of the five numbers is 540, regardless of the individual values.

The mean of five data values multiplied by 5 is the sum of the values, so if Statement 2 is true, then their sum is $108 \cdot 5 = 540$ .

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Example Question #2 : Dsq: Calculating An Angle In A Polygon

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

Statement 1: The polygon has 20 sides.

Statement 2: An exterior angle of the polygon measures $18^{\circ}$ .

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

From Statement 1, you can calculate the measure of an interior angle as follows:

$\frac{180(20-2) }{20} = \frac{180(18) }{20} = 162 ^{\circ }$

From Statement 2, since an interior angle and an exterior angle at the same vertex form a linear pair, they are supplementary, so you can subtract 18 from 180:

$180 - 18 = 162^{\circ }$

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Example Question #11 : Polygons

Is $\textrm{ polygon } ABCDE$ a regular pentagon?

Statement 1: $m \angle A > m \angle B$

Statement 2: CD > DE

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

All of the interior angles of a regular polygon are congruent, as are all of its sides. Statement 1 violates the former condition; statement 2 violates the latter.

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Example Question #4 : Dsq: Calculating An Angle In A Polygon

Thing

Note: Figure NOT drawn to scale.

The diagram above shows a triangle and a rhombus sharing a side. Is that rhombus a square?

Statement 1: The triangle is not equilateral.

Statement 2: $m \angle AVC = 150^{\circ }$

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

To show that the rhombus is a square, you need to demonstrate that one of its angles is a right angle - that is, $90 ^{\circ }$ . Both statements together are insufficent - if $m \angle AVC = 150^{\circ }$ , you would need to demonstrate that $m \angle AVD = 60^{\circ }$ is true or false, and the fact that the triangle is not equilateral is not enough to prove or to disprove this.

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Example Question #4 : Dsq: Calculating An Angle In A Polygon

Hexagon

Note: Figure NOT drawn to scale.

$m \angle A = m \angle C = m \angle E = 119$

What is $m \angle D$ ?

Statement 1: $m \angle B$ , $m \angle D$ , and $m \angle F$ are the first three terms, in order, of an arithmetic sequence.

Statement 2: $m \angle B + 2 = m \angle F$

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

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

Therefore,

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

$\ 119 + m \angle B + 119 + m \angle D + 119 + m \angle F = 720$

$m \angle B + m \angle D + m \angle F + 357 = 720$

$m \angle B + m \angle D + m \angle F = 720 - 357 = 363$

Suppose we only know that $m \angle B$ , $m \angle D$ , and $m \angle F$ are the first three terms of an arithmetic sequence, in order. Then for some common difference ,

$m \angle B = m \angle D - d$

$\ m \angle F = m \angle D + d$

$\ \left (m \angle D - d \right )+ m \angle D \left + ( m \angle D +d \right ) = 363$

$\ 3 \cdot m \angle D \left = 363$

$m \angle D \left = 121$

Suppose we only know that $m \angle B + 2 = m \angle F$ . Then

$m \angle B + m \angle D + \left (m \angle B + 2 \right ) = 363$

$2 \cdot m \angle B + m \angle D + 2 = 363$

$2 \cdot m \angle B + m \angle D = 361$

With no further information, we cannot determine $m \angle D$ .

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Example Question #3 : Dsq: Calculating An Angle In A Polygon

Is the degree measure of an exterior angle of a regular polygon an integer?

Statement 1: The number of sides of the polygon is a factor of 30.

Statement 2: The number of sides of the polygon is a factor of 40.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The sum of the degree measures of the exterior angles, one per vertex, of any polygon is 360, so each exterior angle of a regular polygon with sides measures $\frac{360}{N}$ . Therefore, the measure of one exterior angle of a regular polygon is an integer if and only if is a factor of 360.

If is a factor of a factor of 360, however, then is a factor of 360. 30 and 40 are both factors of 360: $360 \div 30 = 12$ and $360 \div 40 = 9$ . Therefore, it follows from either statement that the number of sides is a factor of 360, and each exterior angle has a degree measure that is an integer.

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Example Question #8 : Dsq: Calculating An Angle In A Polygon

Is the degree measure of an exterior angle of a regular polygon an integer?

Statement 1: The number of sides of the polygon is divisible by 7.

Statement 2: The number of sides of the polygon is divisible by 10.

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

The sum of the degree measures of the exterior angles, one per vertex, of any polygon is 360, so each exterior angle of a regular polygon with sides measures $\frac{360}{N}$ .

For $\frac{360}{N}$ to be an integer, every factor of must be a factor of 360. This does not happen if 7 is a factor of , so Statement 1 disproves this. This may or may not happen if 10 is a factor of - $\frac{360}{40} = 9$ , but $\frac{360}{80} = 4.5$ , so Statement 2 does not provide an answer.

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Example Question #11 : Dsq: Calculating An Angle In A Polygon

A nonagon is a nine-sided polygon.

Is Nonagon ABCDEFGHI regular?

Statement 1: $m \angle A = 150 ^{\circ }$

Statement 2: $m \angle G = 150 ^{\circ }$

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Each angle of a regular nonagon measures $\left [\frac{180 (9-2) }{9 } \right ]^ {\circ } = 140^ {\circ }$

Therefore, each of the two statements proves that the nonagon is not regular.

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Example Question #14 : Polygons

What is the measure of $\angle 1$ ?

Statement 1: $\angle 1$ is an exterior angle of an equilateral triangle.

Statement 2: $\angle 1$ is an interior angle of a regular hexagon.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

An exterior angle of an equilateral triangle measures $360 ^{\circ } \div 3 = 120 ^{\circ }$ . An interior angle of a regular hexagon measures $\left [180 (6-2) \div 6 \right ]^ {\circ} = 120 ^ {\circ}$ . Either statement is sufficient.

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GMAT Math : Polygons

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #2 : Dsq: Calculating An Angle In A Polygon

Example Question #3 : Dsq: Calculating An Angle In A Polygon

Example Question #2 : Dsq: Calculating An Angle In A Polygon

Example Question #11 : Polygons

Example Question #4 : Dsq: Calculating An Angle In A Polygon

Example Question #4 : Dsq: Calculating An Angle In A Polygon

Example Question #3 : Dsq: Calculating An Angle In A Polygon

Example Question #8 : Dsq: Calculating An Angle In A Polygon

Example Question #11 : Dsq: Calculating An Angle In A Polygon

Example Question #14 : Polygons

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