Download In order for a figure to be considered a polygon, it must

SECTION 3-5
THE POLYGON ANGLE-SUM THEOREMS
In order for a figure to be considered a polygon, it must
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Have at least three straight sides, with no curves or bends in a single side
Have the sides only meet up at their endpoints, and not overlap
Have the inside of the figure be “closed off” from everything outside the segments
EXAMPLES OF NON PLOYGONS
EXAMPLES OF POLYGONS
To name a polygon, simply start at any vertex and follow the sides down to the next
vertex in either direction. Once you have the second vertex, continue going in the same
direction until every vertex is accounted for once and ONLY once
Ex. This polygon can be named
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ABCD
ADCB
BADC
CBAD
Etc.
It CANNOT be named
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ACDB
ADBC
BCAD
ACDC
ABCDA
Etc.
Polygons can be classified into two types, convex polygons and concave polygons
In a convex polygon, there are absolutely no diagonals outside of the polygon without
adding or taking away points or changing previously existing sides.
A diagonal is a segment that connects two points of a polygon together, but is not
inside the polygon itself
A concave polygon DOES have diagonals that are at least partly outside the polygon
itself
Some names for the more basic polygons. The general name (not to be confused with the
name based on the vertices) is based on how many sides a triangle has.
Sides
3
4
5
6
8
9
10
12
n
Name
Triangle
Quadrilateral
Pentagon
Hexagon
Octagon
Nonagon
Decagon
Dodecagon
n-agon
Theorem 3-14
The Polygon Angle-Sum Theorem
The sum of the measures of the angles of an n-gon is (n-2)180
What this theorem essentially means is that if you were to substitute the “n” for however
many sides the polygon has, you would be able to discern how much the measures of all
of the polygon’s angles put together are equal to.
Ex. A “15”-gon has 15 sides
(15-2)180
(13) 180
2340
The sum of all the 15-gon’s angles is 2340 degrees.
Uses:
You know the measures of two of the angles, but are unable to find the third.
Using the Polygon Angle-Sum Theorem, you are able to discern the sum of all of the
measures of the angles put together.
(3-1) x 180
1 x 180
180
You now know that
65+30+x=180
Combine like terms
95+x=180
Subtract 95 from both sides
x=85
m<x=85
Theorem 3-15
Polygon Exterior Angle-Sum Theorem
The sum of the measures of the exterior angles of a polygon, one at each vertex, is
360
An Exterior Angle is an angle that is a linear pair to an angle in a polygon. If you
were to continue one of the polygon’s sides outside of the polygon, the angle created
would be an exterior angle.
An exterior angle in a triangle (Angle 4)
An equilateral polygon is a polygon that has all of its sides congruent to one another
An equiangular polygon has all of its angles congruent to one another
A regular polygon is both of the aforementioned qualities
Review Questions
1.
Give at least 3 names for this polygon
2. Is this polygon concave or convex? Explain.
3. Name the total sum of the measures of the angles in this figure
4. Name the total sum of the measures of the exterior angles of these figures
5. Is this figure a polygon? Why or why not?