Download 6-1 Angles in Polygons.notebook

6­1 Angles in Polygons.notebook
October 05, 2015
6­1: <s in Polygons
Remember the Triangle Sum Theorem? It said the interior angles of a triangle add up to 180 degrees.
We will use this later and apply it to polygons!
We name polygons based on their number of sides:
n=3
n=4
triangle quadrilateral
n=7
n=8
heptagon octagon
n=5
n=6
pentagon
hexagon
n=9
n=10
nonagon
decagon
6­1 Angles in Polygons.notebook
October 05, 2015
and...
More than 12?
Call it an "n­gon"
n = 12
dodecagon
ex) 15 sides would
be called a 15­gon
Diagonals: segments drawn from one vertex of a polygon to another.
(must be drawn to non adjacent vertices)
6­1 Angles in Polygons.notebook
October 05, 2015
So, how do we determine the sum of the interior angles of a polygon?
Start by...
Drawing all possible diagonals from one vertex of the polygon.
By drawing non intersecting diagonals. These diagonals make triangles inside of polygons.
n = 4
n = 5
n = 6
n = 7
n = 8
n = 9
RULE: Number of triangles = n ­ 2
What we find is that the number of triangles inside of a polygon relates to the number of sides.
# triangles = n ­ 2
So, if we know that a triangle has 180 degrees and we know how many triangles are inside of a polygon, we can determine how many degrees are inside by multiplying (n ­ 2)180.
6­1 Angles in Polygons.notebook
Sum interior angles of a polygon can be found by
S = 180(n ­ 2)
n ­ 2 tells us how many triangles can be
drawn with non intersecting diagonals
180 is how many degrees the angles of a
triangle add up to
ex) Find the sum of the interior angles of a hexagon.
What is n?
October 05, 2015
6­1 Angles in Polygons.notebook
October 05, 2015
ex] Find missing angles: 5x
(11x+4)
(11x+4)
5x
First, we will need to find the sum of the interior angles.
Then, we will need to solve for x.
Finally, we can plug in for x and find each angle measure.
Regular Polygons
­ All sides are congruent
­ All int angles are congruent
Our first example of this was an
equiangular, equilateral triangle
Triangle: n = 3
180(n ­ 2) = 180
180 / 3 = 60o
6­1 Angles in Polygons.notebook
October 05, 2015
To find the measure of an interior angle of a regular polygon, take the sum and divide it by the number of angles.
Measure of the int angles of a regular polygon:
180(n ­ 2) n
ex) Find the measure of an interior angle of a regular pentagon.
What is n?
Then, plug into formula
6­1 Angles in Polygons.notebook
October 05, 2015
ex) Find the measure of an interior angle of a regular Octagon.
WHAT IS n???
ex) An interior angle of a regular polygon measures 144 degrees. How many sides does the polygon have?
6­1 Angles in Polygons.notebook
October 05, 2015
ex) A regular polygon has an angle measuring 140 degrees. Find the number of sides of the polygon.
Exterior Angles: All exterior angles of a polygon sum to 360, regardless of how many sides the polygon has.
more below
In a regular polygon the exterior angles are congruent.
So, for a regular polygon you can find the measure of a single exterior angle by dividing 360 by n.
Exterior Angles = 360 n
6­1 Angles in Polygons.notebook
October 05, 2015
ex) Find the sum of the exterior angles of a dodecagon.
ex) Find the measure of an exterior angle of a regular pentagon.
Recap:
What is n?
Sum of interior angles of a polygon:
Measure of interior angles of regular polygon:
Sum of exterior angles of a polygon:
Measure of exterior angles of regular polygon:
6­1 Angles in Polygons.notebook
Homework:
pg 397
1 ­ 23
October 05, 2015