SECTION 3-5

THE POLYGON ANGLE-SUM THEOREMS

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

·  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

· 
ABCD

·  ADCB

·  BADC

·  CBAD

·  Etc.

It CANNOT be named

·  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 / Name
3 / Triangle
4 / Quadrilateral
5 / Pentagon
6 / Hexagon
8 / Octagon
9 / Nonagon
10 / Decagon
12 / Dodecagon
n / 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?