CH. 5-1: POLYGON ANGLES Mr. Schaab’s Geometry Class Our Lady of Providence Jr.-Sr. High School

What is a Polygon?  Polygon – a 2-dimensional closed figure with 3 or more sides and 3 or more vertices (corners).  n-gon – What we call a polygon that could have any number of sides.  Example: a polygon with 31 sides is called a 31-gon.  3-gon is another name for a triangle.  Diagonal – a segment that connects two non- consecutive vertices of a polygon.

Polygon Angle Theorems  Theorem 5.1-Polygon Angle-Sum Theorem: The sum of the measures of the interior angles of a convex n-gon is: (n – 2) ∙ 180  What is the sum of the interior angles of the figure on the right? (8-2)(180)=1080°

Polygon Angle Theorems  Recall: Exterior Angle – Angle formed by the extension of one side of a polygon.  Examples:  Theorem 5.2-Polygon Exterior Angles Theorem – The sum of the measures of the exterior angles of ANY polygon is 360°.

Types of polygons  Convex Polygon – all interior angles are less than 180°. These are the polygons that come to mind first when we think about polygons.  Concave Polygon – At least one interior angle is greater than 180°. In a concave polygon, at least one of the diagonals lies outside the polygon. (the polygon has “caved” in sides!)

Regular Polygons  Regular Polygon: a convex polygon that is both equilateral and equiangular.  Can you draw a triangle that is equilateral but not equiangular? No.  Can you draw a convex quadrilateral that is equilateral but not equiangular? Yes. Example:

Regular Polygons  If we have a regular polygon, how can we find the measure of just ONE interior angle without measuring?  If the sum of its measures is (n-2) ∙ 180 and all the angles are the same measure, we can just divide the sum by the number of angles in our regular polygon!

Regular Polygons  Example: Hexagon ABCDEF is a regular hexagon. Find m ∠ B.  Solution: