B E L L R I N G E RB E L L R I N G E R Four angles of a pentagon are Four angles of a pentagon are

6060, 110, 110, 136, 136, and 74, and 74. Find the . Find the measure of the last angle. measure of the last angle.

What is the measure of one angle What is the measure of one angle in a regular hexagon?in a regular hexagon?

SOLUTIONSSOLUTIONS

SOLUTION #1SOLUTION #1 In a pentagon, there are 5 sides, 5 vertices, In a pentagon, there are 5 sides, 5 vertices,

and 5 angles. and 5 angles. We are given the measure of 4 angles, which We are given the measure of 4 angles, which

in total measure: 60in total measure: 60°° + 110 + 110°° + 136 + 136°° + 74 + 74°° = = 380380°.°.

The sum of all the interior angles in a The sum of all the interior angles in a pentagon can be found using (n-2)180. n=5, pentagon can be found using (n-2)180. n=5, so (5-2)180 = 3(180) = 540°. (See so (5-2)180 = 3(180) = 540°. (See exampleexample))

So, the last angle would be 540 – 380 = 160°So, the last angle would be 540 – 380 = 160°

SOLUTIONSSOLUTIONS

SOLUTION #2SOLUTION #2 First, find the sum of all the angles inside of a First, find the sum of all the angles inside of a

hexagon. (n-2)180 = sum of angles inside of a hexagon. (n-2)180 = sum of angles inside of a polygon.polygon.

So, (6-2) 180 = (4) 180 = 720So, (6-2) 180 = (4) 180 = 720°.°. Since there are six angles in the hexagon, Since there are six angles in the hexagon,

and since they must each be the same and since they must each be the same measure (the hexagon is regular), we can measure (the hexagon is regular), we can divide 720 by 6 or 720/6 = 120°.divide 720 by 6 or 720/6 = 120°.

More about POLYGONS:More about POLYGONS:

What about the angles EXTERIOR of the What about the angles EXTERIOR of the polygon?polygon?EXTERIOR angles of a polygon, are the EXTERIOR angles of a polygon, are the ‘LINEAR PAIRS’ of the INTERIOR ‘LINEAR PAIRS’ of the INTERIOR ANGLES. Observe below:ANGLES. Observe below:

1

2

34

5

6∠1, ∠2, ∠3, ∠4, ∠5, and ∠6 are all EXTERIOR angles of the hexagon. They each form a linear pair with the INTERIOR angles of the figure.

Polygon Exterior Angle TheoremPolygon Exterior Angle Theorem

For any polygon, the sum of the measures For any polygon, the sum of the measures of the exterior angles will equal 360of the exterior angles will equal 360°°

OBSERVE in class OBSERVE in class investigationinvestigation..

For example,

25° 70°

85°

155°

95°

110°

155

110

+ 95

360°

ExampleExample

Suppose you are observing a regular Suppose you are observing a regular nonagon. What is the measure of each nonagon. What is the measure of each exterior angle?exterior angle? We know the sum of the exterior angles for We know the sum of the exterior angles for

any polygon must be 360any polygon must be 360°. °. The nonagon is regular—all the interior The nonagon is regular—all the interior

angles must be congruent. Therefore, all of angles must be congruent. Therefore, all of the exterior angles are congruent.the exterior angles are congruent.

There are 9 sides, 9 interior angles, and 9 There are 9 sides, 9 interior angles, and 9 exterior angles. 360/9 = 40° exterior angles. 360/9 = 40°

How about this?How about this?Suppose you have an ‘equiangular’ 1000-gon. Suppose you have an ‘equiangular’ 1000-gon. What is the sum of all the angles in a 1000-gon?What is the sum of all the angles in a 1000-gon? (1000-2)180 = 179,640(1000-2)180 = 179,640°°

What about the measure of one angle in the 1000-gon?What about the measure of one angle in the 1000-gon?179,640/1000 = 179.64179,640/1000 = 179.64°°

What is therefore the measure of one exterior angle? What is therefore the measure of one exterior angle? Since it is a linear pair to the interior angle, Since it is a linear pair to the interior angle, 180 – 179.64 = .360180 – 179.64 = .360°°

So what is the measure of all the exterior angles in a So what is the measure of all the exterior angles in a 1000-gon?1000-gon?

(.360)1000 = 360(.360)1000 = 360° --just like every other polygon that ° --just like every other polygon that has ever existed.has ever existed.

Equiangular Polygon ConjectureEquiangular Polygon Conjecture

In the Bell Ringer, we had found the measure of In the Bell Ringer, we had found the measure of one angle inside of an equiangular polygon one angle inside of an equiangular polygon using our own formula—using our own formula—

(n-2)180 / n (That is, the sum of the angles (n-2)180 / n (That is, the sum of the angles in in the polygon, divided by the number of the polygon, divided by the number of sides/angles in the polygon)sides/angles in the polygon)Now we know a new way to find this information. Now we know a new way to find this information. For any equiangular polygon, the measure of For any equiangular polygon, the measure of just one angle in the polygon can be found by just one angle in the polygon can be found by 180-(360/n)180-(360/n)

The measure of one exterior angle.

Since it’s a linear pair with the interior angle, simply subtract it from 180, and you have the INTERIOR angle!

HUH?HUH?

This means that:This means that: (n-2)180(n-2)180 = 180 – = 180 – 360360 Are these two really the same??Are these two really the same??

n nn n

Let’s see– multiply both sides by ‘n’.Let’s see– multiply both sides by ‘n’.

(n-2)180 = 180n – 360(n-2)180 = 180n – 360

180n – 360 = 180n – 360 (distributive property)180n – 360 = 180n – 360 (distributive property)

Both formulas are the same □Both formulas are the same □

Try This . . .Try This . . .

x

x

x

y

y

135°

40° 50°

x = ?

y = ?

First, you can find y because it is a linear pair with 50°.

y + 50° = 180°

-50 -50

y = 130°

Now, using polygon angle sum we can find x. One of the angles of this septagon we can find since it is a linear pair with 40°. 180 – 40 = 140°

Now use Polygon angle sum. A septagon has 7 sides. (7-2)180. (5)180 = 900°.

So, 130 + 130 + 135 + 140 + x + x + x = 900.

535 + 3x = 900 3x = 365 x = 365/3 = 121.7°

Or This . . . Or This . . .

y

xx

x

116°

82°z

First, y and 116° are LP. 180 -116 = 64°

All of the exterior angles must measure a total of 360°.

Thus, 64 + 82 + x + x + x + 90 = 360°

236 + 3x = 360

-236 -236

3x = 124

x = 124 / 3 = 41.3°

Now find z. It is a Linear Pair with x which we found to be 41.3°

So, z = 180 – 41.3 = 138.7°

CLASSWORKCLASSWORK

P.257,258, #3, 4, 5, 8, 10, 13P.257,258, #3, 4, 5, 8, 10, 13

P. 262 #2-7P. 262 #2-7

Due when you leaveDue when you leave