3.5 Angles of Polygons
In Theorem 3.10, we proved that the sum of the measures of the angles of a triangle is 180. A triangle is one form of polygon, or a many-sided figure, and there are other figures with many sides. A polygon is formed by a collection of segments, each of which only intersects two other segments and only at their endpoints. Polygons can have any number of sides but must have at least three. Polygons will also have any number of interior angles (the same number as sides of the polygon). A convex polygon is one in which no line containing a side of the polygon will intersect the interior of the polygon. Another way to say this is that given a line that contains a side of the polygon, all points of the polygon must be on the same side of that line.
A polygon is a regular polygon if it is convex and all its sides are congruent and all its angles are congruent. An equilateral triangle is an example of a regular polygon. An isosceles triangle is, in general, not regular.
A diagonal of a polygon is a segment that joins nonconsecutive vertices of the polygon. If you were to draw all of the diagonals of a polygon from a single vertex, you form a number of triangles. The number of triangles will be two fewer than the number of sides of the polygon. By determining the sum of all of the interior angles of the triangles, you can find the sum of all of the interior angles of the polygon. For example, if the convex polygon has 4 sides, then you can form 4 minus 2, or 2, triangles. The sum of the angles of each of these two triangles is 180; therefore the sum of all of these interior angles would be 2 times 180 or 360. Again, if the polygon had 5 sides, then the sum of the interior angles would be $(5-2)\cdot 180$, which is equal to 540.
In the drawing below, using the segment tool, , draw the diagonals from each vertex, $A,\;A_1,\;A_2,\;A_3,\;A_4, \text{ and }\;A_5$, to the nonconsecutive vertices of each polygon. How many triangles did you create for each polygon?
This leads us to the following theorem about the sum of the angles of a convex polygon.
- Theorem 3.14:
The sum of the measures of the interior angles of a convex polygon
with $n$ sides is $(n-2)\cdot 180$.
- Theorem 3.15: The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360.
To prove this theorem, we follow the following line of reasoning. At each vertex of the polygon, the sum of the interior angle and its corresponding exterior angle is 180. If the polygon has $n$ sides, then
$$180\cdot n = \text{ sum of all interior angles plus the sum of all of the exterior angles.}$$
Since the sum of the interior angles has been established to be $(n-2)\cdot 180$, then
$$180\cdot n=(n-2)\cdot 180 + \text{ (the sum of the exterior angles).}$$
This leads us to
$$180n = 180n – 360 + \text{ (the sum of the exterior angles)}$$
or
$$360 = \text{ the sum of the exterior angles,}$$
no matter how many sides the polygon has.