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

54 0^{\circ} .

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) 18 0^{\circ}

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

54 0^{\circ} .

Now we are asked to find the relationship between the

54 0^{\circ}

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) 18 0^{\circ}

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.

(x - 2) 18 0^{\circ}

Substitute

$x = x + 1$

(x + 1 - 2) 18 0^{\circ}

CommutativePropAdd

Commutative Property of Addition

(1 + x - 2) 18 0^{\circ}

Distr

Distribute $18 0^{\circ}$

1 \cdot 18 0^{\circ} + (x - 2) 18 0^{\circ}

IdPropMult

Identity Property of Multiplication

18 0^{\circ} + (x - 2) 18 0^{\circ}

Increasing the number of sides of a convex polygon by

1

increases the sum of the interior angle measures by

18 0^{\circ} .

Since

54 0^{\circ}

is

3

times

18 0^{\circ},

we again see that we have to add

3

sides to the polygon to increase the sum of the interior angle measures by

54 0^{\circ} .

It also allows us to write a relation between the answer

3

and the

54 0^{\circ}

increase.

54 0^{\circ} = 3 \cdot 18 0^{\circ}

Exercise