6-1 ANGLES OF A POLYGON

POLYGON: A MANY ANGLED SHAPE

Sides Name

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

8 Octagon

10 Decagon

n n-gon

# sides = # angles = #vertices

SOME INFO:

Regular Polygon: all angles are equal Diagonal: a segment connecting 2

nonconsecutive vertices.

DIAGONALS (Look at these, don’t write in notes) Quadrilateral

Look! 2 triangles 2(180) = 360 Sum of the angles of a quadrilateral is 360

Pentagon 3 triangles 3(180) = 540 Sum of the angles of a pentagon is 540

What do you think about a hexagon? 4(180) = 720

SO . . . . . . . .

THEOREM

The sum of the measures of the INTERIOR angles with n sides is (n – 2)180

The sum of the measures of the exterior angles of any polygon is 360.

ALWAYS 360!!

TWAP—(TRY WITH A PARTNER) HINT: JUST PLUG INTO THE FORMULA! Find a) the sum of the interior angles and

b) the sum of the exterior angles for each shape

1) 32-gon 2) Decagon

Answers:1)a) 5400 b) 3602)a) 1440 b) 360

Other Formulas…

The measure of EACH EXTERIOR angle of a regular polygon is: 360

n(It’s 360 divided by the number of

sides)

The measure of EACH INTERIOR angle of a polygon is: (n-2)180 n

(It’s the SUM of Interior divided by # of sides)

Example

Find the measure of EACH interior angle of a polygon with 5 sides.

(5-2)180 53(180)=540540/5 = 108

EXAMPLE Find the measure of each interior angle of parallelogram RSTU.

Since the sum of the measures of the interior angles is

Step 1 Find the sum of the degrees!

EXAMPLE CONT.Sum of measures of interior angles

EXAMPLE CONTStep 2 Use the value of x to find the measure of each angle.

Answer: mR = 55, mS = 125, mT = 55, mU = 125

mR = 5x= 5(11)= 55

mS = 11x + 4= 11(11) + 4 = 125

mT = 5x= 5(11)= 55

mU = 11x + 4= 11(11) + 4 = 125

To Find # of sides…

Formula: ____360____ 1 ext. angle(360 divided by 1 ext angle)

Also: 1 interior angle + 1 exterior angle = 180

Example

How many sides does a regular polygon have if each exterior angle measures 45º?

360 45 n = 8 sides

EXAMPLE

How many sides does a regular polygon have if each interior angle measures 120º?

Find ext angle: 180-120= 60

360 60

n = 6 sides

EXAMPLE Find the value of x in the diagram.

How many degrees will it =?

Answer: x = 12

5x + (4x – 6) + (5x – 5) + (4x + 3) + (6x – 12) + (2x + 3) +

(5x + 5) = 360

(5x + 4x + 5x + 4x + 6x + 2x + 5x) + [(–6) + (–5) + 3 + (–12) + 3 + 5] = 360

31x – 12 = 360

31x = 372

x = 12

EQUATIONS TO KNOW (FLASHCARDS!!!!)

Sum of interior angles

Each interior angle

Sum of exterior angles

Each exterior angle

# of Sides

2 180n

2 180n

n

360360

n360

1 .Ext

HOMEWORK

Pg. 398 #13-37 odd, 49