In the previous exercise we made a conjecture about the sum of the measures of the angles

S

in an

n

-sided polygon.

S = (n - 2) \cdot 180

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.

S = (n - 2) \cdot 180

▼

Substitute

7

for

n

and evaluate

Substitute

$n = 7$

S = (7 - 2) \cdot 180

SubTerm

Subtract term

S = 5 \cdot 180

Multiply

S = 900

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 .

5 \cdot 180 = 900

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.

S = (n - 2) \cdot 180

▼

Substitute

8

for

n

and evaluate

Substitute

$n = 8$

S = (8 - 2) \cdot 180

SubTerm

Subtract term

S = 6 \cdot 180

Multiply

S = 1080

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 .

6 \cdot 180 = 1080

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.

S = (n - 2) \cdot 180

▼

Substitute

9

for

n

and evaluate

Substitute

$n = 9$

S = (9 - 2) \cdot 180

SubTerm

Subtract term

S = 7 \cdot 180

Multiply

S = 1260

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 .

7 \cdot 180 = 1260

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.

Exercise