Video Transcript
In the figure, if the measure of angle 𝐴 equals 137 degrees, the measure of angle 𝐵
equals 78 degrees, the measure of angle 𝐶 equals 113 degrees, and the measure of
angle 𝐸 equals 131 degrees, find the measure of angle 𝐷.
The shape in the figure is a polygon. More specifically, as it has five sides, it is a pentagon. We’ve been given the measures of four of this pentagon’s interior angles. The measure of angle 𝐴 is 137 degrees, the measure of angle 𝐵 is 78 degrees, the
measure of angle 𝐶 is 113 degrees, and the measure of angle 𝐸 is 131 degrees. We are asked to calculate the measure of the final interior angle of this pentagon,
angle 𝐷.
We can recall the formula for the sum of the interior angle measures in any
polygon. For a polygon with 𝑛 sides, the sum of its interior angle measures, which we denote
as 𝑆 sub 𝑛, is equal to 𝑛 minus two multiplied by 180 degrees. This is derived from the fact that any 𝑛-sided polygon can be divided into 𝑛 minus
two triangles. The sum of the interior angles in a triangle is 180 degrees. So the sum of the interior angles in an 𝑛-sided polygon is 𝑛 minus two lots of 180
degrees.
In this example, in which we have a pentagon, the number of sides is five, and hence
this is the value of 𝑛 in the formula. Hence, the sum of the interior angle measures is equal to five minus two multiplied
by 180 degrees. That’s three multiplied by 180 degrees, which is 540 degrees.
We can now form an equation by summing the measures of all five angles and setting
this expression equal to 540 degrees. Simplifying on the left-hand side gives that the measure of angle 𝐷 plus 459 degrees
is equal to 540 degrees. The measure of angle 𝐷 is therefore equal to 540 degrees minus 459 degrees, which is
81 degrees. This seems reasonable, as from the figure, angle 𝐷 appears to be an acute angle.
So, by recalling the general formula for calculating the sum of the interior angle
measures in any polygon, we’ve determined that the measure of angle 𝐷 is 81
degrees.
