Transcript Section3.4
3.4: The Polygon Angle-Sum Theorem Objective: To classify polygons, and to find the sums of interior and exterior angles of polygons GOOD MORNING!!! It’s going to be a good day Polygon: • • • • Closed plane figure At least 3 sides that are segments Sides intersect only at endpoints Adjacent sides are NOT collinear POLYGONS NOT POLYGONS TO NAME A POLYGON: • Start at any vertex • List vertices consecutively Name the polygon. Name the sides and angles. B A C E D Common Polygons: SIDES NAME 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon n n- gon Convex vs. Concave CONVEX POLYGON • No diagonal with points outside polygon CONCAVE POLYGON • At least one diagonal outside polygon POLYGON-ANGLE-SUM THEOREM The sum of the measures of angles in an n-gon : (n-2)180 Example: Find the sum of the measures of a 15-gon. Finding missing angle measures • Figure out what the sum of the interior angles should be • Add up expressions and set them equal to the sum from part one. EXAMPLE: Find y: y° 120° 85° 53° Example: 1. The sum of the measures of a given polygon is 720. How many sides are in the polygon? 2.Pentagon ABCDE has 5 congruent angles. Find the measure of each angle. Polygon-Exterior-Angle Sum Theorem The sum of the measures of exterior angles of a polygon is ALWAYS 360⁰. Example: What is the sum of exterior angles for a 500-gon? EQUILATERAL POLYGON: All sides EQUIANGULAR POLYGON: All angles REGULAR POLYGON: equilateral and equiangular ONE Interior Measure: ( n 2)180 n measure: ONE Exterior 360 n Examples: 1. The sum of the measures of angles of a polygon with n sides is 1980. Find n, the number of sides. 2. Find the measures of one interior angle and one exterior angle of a REGULAR hexagon. 3. The measure of an exterior angle of a regular polygon is 30⁰. What is the measure of an interior angle? How many sides are there? TYPE 2: Write and answer the following on a separate piece of paper. (2 points each question) 1. Find the sum of the interior and exterior angles of an octagon. 2. Find the measure of one interior angle and one exterior angle of a REGULAR 15-gon. 3. The sum of the interior angles of a polygon is 4680°. Find the number of sides.