Download Angles in Polygons - Mathematical sciences

Triangle Sum Theorem
Draw a Triangle
Make sure your lines are dark!
Tear off two vertices….
Line up the 3 angles (all vertices
touching)
What do they make?
A straight line = 180°
Angle sum of a Triangle


180
<1 + <2 + <3 = 180
2
ALWAYS!!!
1
3
Consider a Quadrilateral

What is the angle sum?
<1 + <2 + <3 + <4 = ?
Quadrilateral

Draw a diagonal…what do you get?
2
3
5
1
4
Two triangles
6
Quadrilateral

Each triangle = 180
2
3
180
1
4
Therefore the two
triangles together =
360
5
180
6
Angle sum of a Quadrilateral

180 + 180 =
360
Consider a Pentagon

What is the angle sum?
Pentagon

Draw the diagonals from 1 vertex
How many triangles?
Angle sum of a Pentagon

Draw the diagonals from 1 vertex
180
180
180
Continue this process through
Decagon

Draw the diagonals from 1 vertex
Continue this process through
Decagon

Draw the diagonals from 1 vertex
What about a 52-gon?


What is
the angle
sum?
Sorry I
can’t draw
it.
Can you
find the
pattern?
Number of
sides
Number of
triangles
Angle sum
of polygon
3
1
180
4
2
360
5
3
6
7
8
Find
the nth
term
Number of
sides
Number of
triangles
Angle sum
of polygon
3
1
180
4
2
360
5
3
6
7
8
9
10
nth
n
Exterior angle sum

Now that you
can find the
angle sum of a
polygon, what
about the
exterior angle
sum?
35
80
65
Exterior angle sum

Note: Extend
each side of the
triangle. This
makes a LINEAR
PAIR
100
80
145
35
65
115
Exterior angle sum



Add up the
exterior angles
100+145+115=
360
100
80
145
35
65
115
Quadrilateral

100+80+55+125 = 360
125
100
80
55
100
80
125
55
Pentagon
72
108
72
108
108
72
72

72 * 5 = 360
108
108
72
What conclusion can you
come up with regarding the
exterior angle sum of a
CONVEX polygon??
The exterior angle sum of a
CONVEX polygon =

360
Interior Angle Measure of a
REGULAR polygons
60
Equilateral Triangle
Angle measure = 60
90
Square
Angle measure = 90
These are measurement that we generally know at this time,
But what about the other regular polygons?
How do we calculate the interior angle measure?
Interior Angle Measure of a
REGULAR polygons
72
120
Calculate by:
Angle Sum
Number of sides
135