### All ISEE Upper Level Math Resources

## Example Questions

### Example Question #1 : How To Find An Angle In Other Polygons

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

**Possible Answers:**

**Correct 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:

### Example Question #2 : How To Find An Angle In Other Polygons

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

**Possible Answers:**

**Correct 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:

### Example Question #2 : How To Find An Angle In Other Polygons

Note: Figure NOT drawn to scale.

Refer to the above diagram. Pentagon is regular. What is the measure of ?

**Possible Answers:**

**Correct answer:**

The answer can be more clearly seen by extending to a ray :

Note that angles have been newly numbered.

and are exterior angles of a (five-sided) regular pentagon in relation to two parallel lines, so each has a measure of . is a corresponding angle to , so its measure is also .

By angle addition,

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

In the above figure, the seven-side polygon, or *heptagon*, shown is regular. What is the measure of ?

**Possible Answers:**

The correct answer is not given among the other responses.

**Correct 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.

is an interior angle of a regular heptagon and therefore has measure

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

is supplementary to , so

### Example Question #6 : Other Polygons

In the above figure, the seven-side polygon, or *heptagon*, shown is regular. What is the measure of ?

**Possible Answers:**

The correct answer is not given among the other responses.

**Correct 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.

and are exterior angles of a (seven-sided) regular heptagon, so each has a measure of . is a corresponding angle to in relation to two parallel lines, so its measure is also .

By angle addition,

### Example Question #7 : Other Polygons

Note: Figure NOT drawn to scale.

In the above figure, Pentagon is regular. Give the measure of .

**Possible Answers:**

**Correct answer:**

The sum of the degree measures of the angles of Quadrilateral is 360, so

Each interior angle of a regular pentagon measures

,

which is therefore the measure of .

It is also given that and , so substitute and solve:

### Example Question #1 : Other Polygons

Note: Figure NOT drawn to scale.

In the above figure, Pentagon is regular. Give the measure of .

**Possible Answers:**

The correct answer is not given among the other responses.

**Correct answer:**

The sum of the degree measures of the angles of Quadrilateral is 360, so

.

Each interior angle of a regular pentagon measures

,

which is therefore the measure of both and .

and form a linear pair, making them supplementary. Since ,

.

Substitute and solve:

### Example Question #21 : Plane Geometry

The measures of the angles of an octagon form an arithmetic sequence. The greatest of the eight degree measures is . What is the least of the eight degree measures?

**Possible Answers:**

This octagon cannot exist.

**Correct answer:**

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

.

In an arithmetic sequence, the terms are separated by a common difference, which we will call . Since the greatest of the degree measures is , the measures of the angles are

Their sum is

The least of the angle measures is

The correct choice is .

### Example Question #22 : Plane Geometry

What is of the total number of degrees in a 9-sided polygon?

**Possible Answers:**

**Correct 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:

Therefore, of the total sum of degrees in a 9 sided polygon would be equal to 180 degrees.

### Example Question #23 : Plane 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 . What is the greatest of the nine degree measures?

**Possible Answers:**

**Correct answer:**

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

.

In an arithmetic sequence, the terms are separated by a common difference, which we will call . Since the least of the degree measures is , the measures of the angles are

Their sum is

The greatest of the angle measures, in degrees, is

is the correct choice.