A pentagon has two exterior angles that measure (3x)°, two exterior angles that measure (2x+22)°, and an exterior angle that measures (x+41)°. If all of these angles have different vertices, what is the measure of the smallest exterior angle?
25°25 degrees
72°72 degrees
75°75 degrees
66°
since the exterior angles add up to 360,
2(3x) + 2(2x+22) + x+41 = 360
Now just solve for x and find which of the three values is smallest.
Extra credit: what is the smallest interior angle?
To find the measure of the smallest exterior angle of the pentagon, we need to determine the values of x and then substitute the values into the expressions for the exterior angles.
Let's go step by step to solve the problem:
1. We know that the sum of the exterior angles of any polygon is always 360 degrees. Hence, we have the equation:
(3x) + (2x+22) + (2x+22) + (x+41) + (smallest exterior angle) = 360
2. Next, simplify the equation:
8x + 85 + (smallest exterior angle) = 360
3. Rearrange the equation:
smallest exterior angle = 360 - (8x + 85)
4. To find the smallest exterior angle, substitute the given answer choices for x and calculate the corresponding value for the smallest exterior angle.
Let's substitute the first answer choice, 25.
smallest exterior angle = 360 - (8 * 25 + 85)
smallest exterior angle = 360 - (200 + 85)
smallest exterior angle = 360 - 285
smallest exterior angle = 75
Therefore, the measure of the smallest exterior angle is 75°. Hence, the correct answer is:
75°75 degrees