The measures of the interior angles of a pentagon satisfy the ratio three to four to four to four to five. What is the measure of the smallest angle?
There are a few things we need to sort out. First, we need to know the measure of all of the interior angles inside a pentagon. This is something you might remember. But if you don’t, there’s a formula. To find the sum of interior angles in a polygon with 𝑛 sides. We take the number of sides in that polygon, subtract two, and then multiply by 180 degrees. We’re dealing with a pentagon. Which means we’re dealing with a five-sided polygon. Five minus two is three. And three times 180 is 540 degrees. And so, for our pentagon, the sum of the interior angles is 540 degrees.
There are five angles inside this pentagon. And they are in the ratio three to four to four to four to five. This means three of the angles have the same measure. How can we go from this ratio to finding out the actual value of all five of these angles? The ratio is from angle to angle to angle to angle to angle. And we know what the sum of all these angles should be. We could take this ratio and find the sum of all the angles in the ratio. We could say three plus four plus four plus four plus five equals 20. And then we would have three to four to four to four to five to 20.
We’re interested in the smallest angle out of 540 degrees. And it will be in the ratio of three to 20. We know that 20 multiplied by some value would equal 540. And to find that, we need to divide 540 by 20. 20 can be divided into 54 two times. Two times 20 equals 40. 54 minus 40 equals 14. Bring down the zero. And then we need to divide 20 into 140. I know that seven times two equals 14. And that means seven times 20 equals 140.
And so, we can say that 20 times 27 equals 540. And if we multiply our denominator by 27, then we have to multiply our numerator by 27. Three times 27 equals 81. We’re dealing with angle measures, so that’s degrees. The smallest angle compared to the sum of all the angles is 81 degrees out of 540 degrees. Our question is only interested in the measure of the smallest angle, which is then 81 degrees.