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Polygons
Convex vs. Concave Polygons
Interior Angles of Polygons
Exterior Angles of Polygons
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.
A rose by any other name…
 To name a polygon, start at a vertex and
either go clockwise or counterclockwise.
a
b
c
f
e
d
Diagonals
 A diagonal of a polygon is any segment that
connects two nonconsecutive (nonadjacent)
vertices of the polygon.
Convex polygons

A polygon in which each interior angle has a
measure less than 180.
Polygons can be CONCAVE or CONVEX
CONCAVE
CONVEX
Classify each polygon as convex or
concave.
Octagon
Triangle
Nonagon
Quadrilateral
Decagon
Pentagon
Dodecagon
Hexagon
Heptagon
15 sides Pentadecagon
n-gon
Important Terms
EQUILATERAL - All sides are congruent
EQUIANGULAR - All angles are congruent
REGULAR - 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
4
4(180) = 720
n-2
(n-2) 180
6
n
Regular Polygons
No. of sides
Name
Angle Sum
Interior Angle
3
triangle
180°
60°
4
quadrilateral
360°
90°
5
pentagon
540°
108°
6
hexagon
720°
120°
7
heptagon
900°
128 7/9°
8
octagon
1080°
135°
9
nonagon
1260°
140°
10
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.
540°
B)Find the measure of
ONE interior angle
108°
Exterior angles of a triangle
A
interior
opposite
angles
B
exterior angle
C
i.e. ACD = ABC + BAC
D
The exterior angle of a
triangle is equal to
the sum of the interior
opposite angles.
Example
55°
A
85°
35°
C
20°
40°
60°
120°
120°
20°
E
40°
D
Find
CED = 120°
ACE = 35°
CDE = 40°
ABE = 20°
EAB = 40°
AEB = 120°
CAE = 85°
B
Two more important terms
Interior
Angles
Exterior
Angles
Exterior angles of a polygon
a
b
Exterior angles of a
polygon add to 360°.
c
e
d
a + b + c + d + e = 360°
At each vertex: interior angle + exterior angle = 180°
In any convex
polygon, the sum of
the measures of the
exterior angles, one at
each vertex, is 360°.
2
1
3
5
4
m1  m2  m3  m4  m5  360
o
In any convex
polygon, the sum of
the measures of the
exterior angles, one at
each vertex, is 360°.
1
3
2
m1  m2  m3  360
o
In any convex
polygon, the sum of
the measures of the
exterior angles, one at
each vertex, is 360°.
1
2
4
3
m1  m2  m3  m4  360
o
Find the measure of ONE exterior
angle of a regular hexagon.
sum of the exterior angles

number of sides
360

6
o
60°
Find the measure of
ONE exterior angle of a regular heptagon.
sum of the exterior angles

number of sides
360

7
o
51.4°
Each exterior angle of a polygon is 18.
How many sides does it have?
sum of the exterior angles
 exterior angle
number of sides

360
 18
n
n = 20
The sum of the measures of five interior
angles of a hexagon is 535o.
What is the measure of the sixth angle?
185°
The measure of the exterior angle of a
quadrilateral are x, 3x, 5x, and 3x.
Find the measure of each angle.
x + 3x + 5x + 3x = 360o
12x = 360o
x = 30o
Use substitution to solve for each angle measure.
30°, 90°, 150°, and 90°
If each interior angle
of a regular polygon is 150,
then how many sides
does the polygon have?
n = 12
Example
B
C
ABCDE is a
regular hexagon
with centre O.
120°
A
30°
30°
60°
O
F
Find
60° 60°
D
E
ABC = 120°
ACD = 90°
ADC = 60°
BAC = 30°
ODE = 60°
CAD = 30°
EOD = 60°