If four angles of a convex pentagon measure 120°, 93°, 108°, and 101°, find the measure of the fifth angle. Show equations and work that lead to your answer.
The sum of all internal angles of a pentagon is 540 degrees.
For a polygon of N sides, the sum is
N*(180-360/N) = N*180 - 360 degrees
Subtract (120 +93 +108 +101) from 540 for your answer.
To find the measure of the fifth angle in a convex pentagon, we can use the fact that the sum of the angles in any polygon is given by the formula:
Sum of angles = (n - 2) * 180 degrees
where n is the number of sides of the polygon. In the case of a pentagon, n = 5. Therefore, the sum of the angles in a pentagon can be found using:
Sum of angles = (5 - 2) * 180 degrees = 540 degrees.
We know the measures of four of the angles in the pentagon: 120°, 93°, 108°, and 101°. Let's denote the measure of the fifth angle as x.
To find x, we can subtract the sum of the four known angles from the sum of angles in the pentagon:
x + 120° + 93° + 108° + 101° = 540°.
Simplifying the equation:
x + 422° = 540°.
To isolate x, we can subtract 422° from both sides of the equation:
x = 540° - 422°.
Simplifying:
x = 118°.
Therefore, the measure of the fifth angle in the convex pentagon is 118°.