Exercise 9 Page 66

In the previous exercise we made a conjecture about the sum of the measures of the angles $S$ in an $n$-sided polygon. Let's test it on other polygons. We can try it on a heptagon, an octagon, and a nonagon.

Heptagon

A heptagon is a $7$-sided polygon. Therefore, we will substitute $n=7$ in our formula. According to our conjecture, the sum of the measures of the angles in a heptagon is $900.$ Let's draw it and calculate this sum without the formula. We divided the heptagon into $5$ triangles. The sum of the measures of the angles in each of them is $180.$ Therefore, the sum of the measures of the angles in the heptagon is $5$ times $180.$ As we can see, it is the same as the sum we got using our conjecture, so we can tell that it works in this case.

Octagon

An octagon is an $8$-sided polygon. Therefore, we will substitute $n=8$ in our formula. Now, let's draw it and calculate this sum without the formula. We divided the octagon into $6$ triangles. The sum of the measures of the angles in each of them is $180.$ Therefore, the sum of the measures of the angles in the octagon is $6$ times $180.$ It is the same as the sum we got using our conjecture, so we can tell that it also works in this case.

Nonagon

A nonagon is a $9$-sided polygon. Therefore, we will substitute $n=9$ in our formula. Now, let's draw it and calculate this sum without the formula. As we can see, it divides into $7$ triangles. The sum of the measures of the angles in each of them is $180.$ Therefore, the sum of the measures of the angles in the nonagon is $7$ times $180.$ It is the same as the sum we got using our conjecture, so we can tell that it also works in this case.

Conclusion

Our conjecture worked for each polygon we tried it on. Therefore, we can conclude that it holds true.