# GMAT Math : Polygons

## Example Questions

### Example Question #141 : Geometry

What is the sum of six numbers,  ?

Statement 1:  are the measures of the interior angles of a hexagon.

Statement 2: The median of the data set  is 120.

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.

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.

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 , 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.

### Example Question #1 : Dsq: Calculating An Angle In A Polygon

What is the sum of five numbers,  ?

Statement 1:  are the measures of the interior angles of a pentagon.

Statement 2: The mean of the data set  is 108.

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.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The sum of the measures of the five interior angles of any pentagon is , 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 .

### Example Question #1 : 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 .

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.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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

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:

### Example Question #6 : Dsq: Calculating An Angle In A Polygon

Is  a regular pentagon?

Statement 1:

Statement 2:

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.

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.

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.

### Example Question #7 : Dsq: Calculating An Angle In A Polygon

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:

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.

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, . Both statements together are insufficent - if , you would need to demonstrate that  is true or false, and the fact that the triangle is not equilateral is not enough to prove or to disprove this.

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

Note: Figure NOT drawn to scale.

What is  ?

Statement 1: , and  are the first three terms, in order, of an arithmetic sequence.

Statement 2:

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.

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.

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

Therefore,

Suppose we only know that , and  are the first three terms of an arithmetic sequence, in order. Then for some common difference

Suppose we only know that . Then

With no further information, we cannot determine .

### Example Question #1 : 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.

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.

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 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.

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 . 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:  and . 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.

### Example Question #1 : 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.

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.

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.

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

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

For  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  - , but , so Statement 2 does not provide an answer.

### Example Question #151 : Geometry

A nonagon is a nine-sided polygon.

Is Nonagon  regular?

Statement 1:

Statement 2:

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.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Each angle of a regular nonagon measures

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

### Example Question #11 : Polygons

What is the measure of   ?

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

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

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

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.

BOTH statements TOGETHER are insufficient to answer the question.