Recall how many sides we had to add in the mentioned exercise. Then, look for a relation between this number and 540^(∘).
540 = 3* 180^(∘)
Practice makes perfect
In the mentioned exercise, we were given a convex polygon and were asked to find the number of sides we would have to add to the polygon to increase the sum of the interior angle measures by 540^(∘). To find the answer, we used the Polygon Interior Angles Theorem, which gives us a formula for the sum of the interior angle measures of a convex polygon with n sides.
(n-2)180^(∘)
Using this formula, we wrote an equation and solved it to find that we would have to add 3 sides to the polygon to increase the sum of the interior angle measures by 540^(∘). Now we are asked to find the relationship between the 540^(∘) increase and the answer. To do so, let's say we have a polygon with x sides. We can find the sum of the interior angle measures using the formula.
( x-2)180^(∘)
Let's see what happens to the sum of the interior angle measures when we increase the number of sides by 1. To do so, we substitute x + 1 for x in our expression.
Increasing the number of sides of a convex polygon by 1 increases the sum of the interior angle measures by 180^(∘). Since 540^(∘) is 3 times 180^(∘), we again see that we have to add 3 sides to the polygon to increase the sum of the interior angle measures by 540^(∘). It also allows us to write a relation between the answer 3 and the 540^(∘) increase.
540^(∘) = 3* 180^(∘)
