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As we discussed before, the three angles of a triangle always add up to 180°.

2 ABC Triangles

In each case m < A + m < B + m < C = 180 degrees . By the way, m < A means "the measurement of angle A".

To find the total number of degrees in any polygon, all we have to do is divide the shape into triangles. To do this start from any vertex and draw diagonals to all non-adjacent vertices.


Here is a quadrilateral.2 Quadrilaterals
If we draw all the diagonals from a vertex we get two triangles.
Each triangle has 180°, so 2 ×180° = 360° in a quadrilateral. 

Pentagon – 5 sidesPentagon3 triangles × 180° = 540°
Hexagon – 6 sidesHexagon4 triangles × 180° = 720°
Septagon – 7 sidesSeptagon5 triangles × 180° = 900°
Octagon – 8 sidesOctagon6 triangles × 180° = 1080°

Are you noticing a pattern? Turns out, the number of triangles formed by drawing the diagonals is two less than the number of sides. If we use the variable n to equal the number of sides, then we could find a formula to calculate the number of degrees in any polygon:

(n-2) x 180 degrees

Example 1

What is the sum of the angles in a dodecagon?dodecagon

A dodecagon has 12 sides, so (12-2) x 180 degrees = 1800 degrees.


Example 2

What is the measure of each angle in a regular nonagon?

nonagon

A nonagon has 9 sides. Using our formula, (9-2) x 180 degrees = 1260 degrees.


Example 3

Find the missing angle.

Triangle (find missing angle)

A triangle has 180°. If we add the measures of angles I and J and subtract from 180, we get:

180 - (35 + 115) degrees


Example 4

Find the measurement of angle Q

hexagon LMNOPQ

This is a hexagon. The total number of degrees equals:
(6-2) x 180 degrees

4 x 180 degrees

720 degrees


Example 5

Find the missing angles in this isosceles trapezoid.

Isosceles Trapezoid

Since this is an isosceles trapezoid, angles I and L are congruent. In addition, angles J and K are congruent, so

m < L = m < I = 45 degrees


Exercise 1

How many degrees are there in a decagon?

Exercise 2

What is the measure of each angle in a regular hexagon?

Exercise 3

Find the missing angle in the triangle.

Missing Angle Triangle (#3)

Exercise 4

Find the missing angle in the triangle.

MIssing Angle Triangle (#4)

Exercise 5

Find the missing angle.

Missing Angle Triangle (#5)

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