Question

A pentagon has two exterior angles that measure (4x), two exterior angles that measure (3x + 76)°, and an exterior angle that measures (3x + 38)°. If all of these angles have different vertices, what is the measure of the smallest exterior angle?

A) 68
B) 10
C) 40
D) 106

Answer 1

To find the measure of the smallest exterior angle of the pentagon, we can set up an equation based on the fact that the sum of the exterior angles of a polygon is always 360 degrees. By solving this equation, we can determine the measure of the smallest exterior angle. The answer is none of the provided options.

Step-by-step explanation:

To find the measure of the smallest exterior angle of the pentagon, we can set up an equation based on the fact that the sum of the exterior angles of a polygon is always 360 degrees.

Let's set up the equation:

1. First exterior angle: 4x
2. Second exterior angle: 3x + 76
3. Third exterior angle: 3x + 38
4. Fourth exterior angle: Unknown
5. Fifth exterior angle: Unknown

Based on the equation, we know that:

4x + 3x + 76 + 3x + 38 + Unknown Angle + Unknown Angle = 360

Combine like terms:

10x + Unknown Angle + Unknown Angle + 114 = 360

Subtract 114 from both sides:

10x + Unknown Angle + Unknown Angle = 246

Since the measure of each exterior angle of a polygon is x degrees, we can set up an equation:

x + Unknown Angle + Unknown Angle = 246

Subtract x from both sides:

Unknown Angle + Unknown Angle = 246 - x

The measure of the smallest exterior angle will be the smallest value for the Unknown Angle, so we want to minimize it. This means we want to maximize the value of x.

Since x will be the greatest angle, we can say:

x ≤ 360/10

x cannot be greater than 36 degrees.

Therefore, the measure of the smallest exterior angle of the pentagon will be the smallest value for the Unknown Angle, which is x.

Therefore, the answer is none of the provided options.