Convex vs. Concave PolygonsInterior Angles of PolygonsExterior Angles of Polygons

Convex vs. Concave PolygonsInterior Angles of PolygonsExterior Angles of Polygons

PolygonsPolygons

To be or not to be…To be or not to be… Polygons consist of entirely segments Consecutive sides can only intersect at

endpoints. Nonconsecutive sides do not intersect.

Vertices must only belong to one angle Consecutive sides must be noncollinear.

Polygons consist of entirely segments Consecutive sides can only intersect at

endpoints. Nonconsecutive sides do not intersect.

Vertices must only belong to one angle Consecutive sides must be noncollinear.

A rose by any other name…

A rose by any other name…

To name a polygon, start at a vertex and either go clockwise or counterclockwise.

To name a polygon, start at a vertex and either go clockwise or counterclockwise. a b

c

de

f

DiagonalsDiagonals

A diagonal of a polygon is any segment that connects two nonconsecutive (nonadjacent) vertices of the polygon.

A diagonal of a polygon is any segment that connects two nonconsecutive (nonadjacent) vertices of the polygon.

Convex polygonsConvex polygonsA polygon in which each interior angle has a

measure less than 180. A polygon in which each interior angle has a

measure less than 180.

Polygons can be CONCAVE or CONVEX

CONVEX

CONCAVE

Classify each polygon as convex or concave.

Classify each polygon as convex or concave.

Triangle

Quadrilateral

Pentagon

Hexagon

Heptagon

Octagon

Nonagon

Decagon

Dodecagon

n-gon

15 sides Pentadecagon

Important TermsImportant Terms

EQUILATERAL - All sides are congruentEQUIANGULAR - All angles are congruentREGULAR - All sides and angles are congruent

# of sides

# of triangles

Sum of measures of interior angles

3 1 1(180) = 180

4 2 2(180) = 360

5 3 3(180) = 540

6 4 4(180) = 720

n n-2 (n-2) 180

Regular PolygonsRegular Polygons

No. of sides Name Angle Sum Interior Angle

3 triangle

4

5

6

7

8

9

10

quadrilateral

180° 60°

360° 90°

pentagon 540° 108°

hexagon 720° 120°

heptagon 900° 128 7/9°

octagon 1080°

135°

nonagon 1260° 140°

decagon 1440°

144°

If a convex polygon has n sides, then the sum of the measure of the interior angles is (n – 2)(180°)

Use the regular pentagon to answer the questions.

A)Find the sum of the measures of the interior angles.

B)Find the measure of ONE interior angle

540°

108°

Exterior angles of a triangleExterior angles of a triangle

The exterior angle of a triangle is equal to the sum of the interior opposite angles.

interior opposite angles

exterior angle

A

B C D

i.e. ACD = ABC + BAC

20°C

A

B

D

E

Find CED

= 40°

40°

CDE

= 40°

40°

EAB

60°

= 120°

120°

55°

CAE= 85°

85°

ACE

35°

= 35°

ABE= 20°

20°

AEB= 120°

120°

Example

Two more important terms

Exterior Angles

Interior Angles

b

Exterior angles of a polygonExterior angles of a polygon

Exterior angles of a polygon add to 360°.

At each vertex: interior angle + exterior angle = 180°

a

c

ea + b + c + d + e = 360°

d

In any convex polygon, the sum of the measures of the exterior angles, one at each vertex, is 360°.

1

2

3

4

5

m1m2 m3m4 m5 360o

In any convex polygon, the sum of the measures of the exterior angles, one at each vertex, is 360°.

1

3

2

m1m2 m3 360o

In any convex polygon, the sum of the measures of the exterior angles, one at each vertex, is 360°.

1

3

2

4

m1m2 m3m4 360o

Find the measure of ONE exterior angle of a regular

hexagon.

Find the measure of ONE exterior angle of a regular

hexagon.

60°

sum of the exterior angles

number of sides

360o

6

Find the measure ofONE exterior angle of a regular

heptagon.

Find the measure ofONE exterior angle of a regular

heptagon.

51.4°

sum of the exterior angles

number of sides

360o

7

Each exterior angle of a polygon is 18. How many sides does it have?

Each exterior angle of a polygon is 18. How many sides does it have?

n = 20

angleexterior sides ofnumber

anglesexterior theof sum

18360

n

The sum of the measures of five interior angles of a hexagon is 535o.

What is the measure of the sixth angle?

The sum of the measures of five interior angles of a hexagon is 535o.

What is the measure of the sixth angle?

185°

x + 3x + 5x + 3x = 360o

12x = 360o

x = 30o

Use substitution to solve for each angle measure.

The measure of the exterior angle of a quadrilateral are x, 3x, 5x, and 3x.

Find the measure of each angle.

The measure of the exterior angle of a quadrilateral are x, 3x, 5x, and 3x.

Find the measure of each angle.

30°, 90°, 150°, and 90°

If each interior angle of a regular polygon is 150,

then how many sides does the polygon have?

If each interior angle of a regular polygon is 150,

then how many sides does the polygon have?

n = 12

Find ABC= 120°

120°

Example

ADC= 60°

60°

BAC= 30°

30°

CAD= 30°

30°

ABCDE is a regular hexagon with centre O.

C

A

B

D

EF

O

ACD

ODE

EOD

= 90°

= 60°

60°

= 60°

60°