GEOMETRY
LESSON 8-1
Pg 404-409
ANGLES OF POLYGONS
Objectives: 1. Find the sum of the INTERIOR Angles of any polygon
2. Find the sum of the EXTERIOR Angles of any polygon
Key Concept: Knowing the sum of the interior angles of a TRIANGLE is important. 180 Degrees
Therefore if we convert ANY polygon into a numerous triangles, we can determine the total sum of the interior
angles of all of the triangles, which is then the sum of the interior angles of the original polygon. Let’s look at
an example of what this means.
Pentagon
Pentagon formed into triangles (3)
Notice the Pentagon (five-sided polygon) forms three triangles. And the sum of the interior angles of each
triangle is 180 degrees, we then have 3(180) = 540 degrees
This is the CONCEPT of THEOREM 8.1 “INTERIOR ANGLE SUM THEOREM”
S = 180(n – 2)
where “S” is the sum of all interior angles of any polygon with “n” sides
Or …S =The measure of a single interior angle (A) times (n) = (A)(n)
Look at the pentagon again: S = 180(n – 2)
S = 180(5 -2)
S = 180(3)
S = 540 degrees
(A)(n) = 180(n – 2)
## WE CAN ALSO USE THIS FORMULA TO DETERMINE THE NUMBER OF SIDES:
(A)(n) = 180(n – 2)
EXAMPLE: If we know that one angle equals 140 degrees, how many sides does the polygon contain?
(A)(n) = 180(n – 2)
140(n) = 180(n – 2)
140n = 180n – 360
360 = 180n – 140n
360 = 40n
9=n
WHAT ABOUT THE SUM OF THE MEASURES OF THE EXTERIOR ANGLES OF ANY
POLYGON?
***Since the exterior angles are always supplement to its associated interior angle, both together must be
equal to 180 degrees.
If we had an equilateral triangle with interior angles of 60 degrees, the corresponding exterior angle
must be 120. Since we have three of those angles 120(3) = 360 degrees. (EXTERIOR SUM)
If we had a square with interior angles of 90 degrees, the corresponding exterior angles must be 90
degrees also. Since we have four of those angles 90(4) = 360 degrees (EXTERIOR SUM)
SEE THE TREND? So for ANY polygon there is, the EXTERIOR ANGLE SUM IS ALWAYS 360
This is THEOREM 8.2 “EXTERIOR ANGLE SUM THEOREM”