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

1

2

3

ALWAYS!!!

Consider a Quadrilateral What is the angle sum?

<1 + <2 + <3 + <4 = ?

Quadrilateral Draw a diagonal…what do you

get?

Two triangles

1

2 3

4

5

6

Quadrilateral Each triangle = 180

Therefore the two triangles together = 360

1

2 3

4

5

6

180

180

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?

65

35

80

Exterior angle sum Note: Extend

each side of the triangle. This makes a LINEAR PAIR

65

35

80100

145

115

Exterior angle sum

Add up the exterior angles

100+145+115=

360

65

35

80100

145

115

Quadrilateral 100+80+55+125 = 360

100 125

80 55

8055

100125

Pentagon

72* 5 = 360

108

108

108

108

108 72

72

72

72

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 90

Equilateral Triangle Angle measure = 60

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

135

Calculate by:

Angle Sum

Number of sides