Angle Measures in Polygons Geometry

Polygon Angle Measures # of sides # of congruent triangles Sum of measures of interior angles TRIANGLE QUADRILATERAL PENTAGON HEXAGON ….. N-GON

POLYGON INTERIOR ANGLES THEOREM The sum of the measures of the interior angles of a convex n-gon is… (n – 2) · 180

Regular Polygons What is the difference between a polygon and a regular polygon?

More Regular Polygons

COROLLARY TO THEOREM 11.1  

Example

Find the value of x in the diagram. Example Find the value of x in the diagram. 114° 105° x° 135° 102°

Example The measure of each interior angle of a regular polygon is 140°. How many sides does the polygon have?

Example The measure of each interior angle of a regular polygon is 165 degrees. How many sides does the polygon have?

Examples A heptagon has four interior angles that measure 160 degrees each and two interior angles that are right angles. What is the measure of the other interior angle? Find the measure of each angle in a regular 11-gon.

Interior vs Exterior

Walk around the polygon Exterior Angles Walk around the polygon

POLYGON EXTERIOR ANGLES THEOREM The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360.

COROLLARY TO THEOREM 11.2  

Example

Find x

Example The measure of each exterior angle of a regular polygon is 40 degrees. How many sides does the polygon have?

Example The sign on a building has a long narrow hexagon shape. Four of the interior angles measure 140 degrees each. The remaining two interior angles at each end of the sign are congruent. What is the measure of each angle.

Example If you were designing a sign for a new building, would it be possible to make a sign that is a regular polygon with each angle having a measure of…. a. 160 degrees b. 115 degrees