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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°
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