Honors Geometry Sections 3.1 & 3.6 Polygons and Their Angle Measures

The word polygon means many sides The word polygon means many sides. In simple terms, a polygon is a many-sided closed figure.

Formally, a polygon is a figure formed from three or more line segments such that each segment intersects exactly two other segments, one at each endpoint, and no two segments with a common endpoint are collinear. The segments are called the_____ of the polygon and the common endpoints are called the _______ of the polygon. sides vertices

When naming a polygon, you must list the vertices in order either clockwise or counterclockwise. The polygon at the right could be named _______ or _______ ABCDEF BAFEDC

A diagonal of a polygon is a segment joining two nonadjacent vertices.

A polygon is equilateral iff A polygon is equiangular iff A polygon that is both equilateral and equiangular is called a _______ polygon. all its sides are congruent. all its angles are congruent. regular

The center of a regular polygon is the point which is equidistant from each of the vertices.

Polygons are classified according to the number of its sides Polygons are classified according to the number of its sides. 3 - ____________ 4 - ____________ 5 - ____________ 6 - ____________ 7 - ____________ 8 - ____________ 9 - ____________ 10 - ___________ 12 - ___________ n - ___________ triangle quadrilateral pentagon hexagon heptagon octagon nonagon decagon dodecagon n - gon

A polygon is convex iff the line containing a side does not contain a point in the interior. A polygon that is not convex is concave.

For each figure, draw all the diagonals from one vertex and complete the table.

Theorem 3.6.1 The sum of the measures of the interior angles of a (convex) polygon with n sides is

Corollary to Theorem 3.6.1 The measure of each interior angle of a regular n-gon is

Example 1: Find the sum of measures of the interior angles of a dodecagon. Example 2: Find the measure of each interior angle of a regular 20-gon.

While the sum of the interior angles of a polygon changes as the number of sides changes, this is not the case with the sum of the exterior angles.

Theorem 3.6.3 The sum of the measures of the exterior angles of a (convex) polygon, one at each vertex, with n sides is

Here’s an example of why that is the case Here’s an example of why that is the case. Adding the five equations together, we get:

Corollary to Theorem 3.6.3 The measure of each exterior angle of a regular n-gon is

Complete this table for regular polygons.

Complete this table for regular polygons.