Properties of Polygons The students will be able to describe the characteristics of a figure and to identify polygons.

Unit G2 Polygon A polygon is a closed plane figure whose sides are segments that intersect only at their endpoints. No two sides with a common endpoint are collinear. Which figure is a polygon? A BC B

Unit G3 The common endpoint of two sides is a vertex of the polygon. are sides of this polygon Polygons Each segment that forms the polygon is a side of the polygon. A E D C B A, B, C, D, E are vertices of this polygon.

Unit G4 Parts of a Polygon Two vertices of a polygon that are connected by a side are called consecutive vertices. Two sides of a polygon that are connected by a vertex are called consecutive sides. These are consecutive vertices. These are consecutive sides A line that connects two non-consecutive vertices is called a diagonal. This is a diagonal

Unit G5 Comparing Polygons You can describe polygons by comparing the lengths of the sides or the measures of the angles. If all the sides of a polygon are con- gruent, then it is an equilateral polygon. If all the angles of a polygon are con- gruent, then it is an equiangular polygon. If a polygon is both equilateral and equi- angular then it is a regular polygon.

Unit G6 Concave or Convex A polygon can be either concave or convex. If you can draw diagonal so that it is outside of the polygon, then the polygon is concave. If all diagonals are inside the polygon, then the polygon is convex. Are these polygons concave or convex? concave convex

Unit G7 Classifying Polygons by the Number of Sides A triangle is a polygon with 3 sides A quadrilateral is a polygon with 4 sides A pentagon is a polygon with 5 sides

Unit G8 A hexagon is a polygon with 6 sides A heptagon is a polygon with 7 sides A octagon is a polygon with 8 sides Classifying Polygons by the Number of Sides

Unit G9 A polygon with 9 sides is a A polygon with 10 sides is a A polygon with 12 sides is a A polygon with n sides is a Classifying Polygons by the Number of Sides

FHSUnit G10 Interior Angles of a Polygon What is the measure of the sum of all the angles of a triangle? If you can divide a polygon into triangles, then you can find out how many degrees are in the polygon. Watch Out! - Every vertex of every triangle that you draw must also be a vertex of the polygon.

FHSUnit G11 Let’s try this method on this pentagon. Interior Angles of a Polygon First, draw the triangles, making sure we draw each line from the same vertex. Next, count the triangles and multiply by 180º This is the measure of the sum of the angles of the polygon.

FHSUnit G12 Polygon Interior Angle Sum Theorem We don’t always want to draw a picture of the polygon and divide it into triangles in order to find the sum of the angles in the polygon. So we have a formula we can use. The formula for the sum of the interior angle measures of a convex polygon with n sides is (n - 2)180º. The (n – 2) portion tells us how many triangles can be drawn in that polygon.

FHSUnit G13 Each Interior Angle If the polygon is regular, then you can find the measure of one interior angle by dividing the sum of the interior angles by the number of sides of the polygon. Example: Find one interior angle of a regular pentagon. First, you find the sum of the interior angles. (n – 2)·180º = (5 – 2)·180º = 3·180º = 540º Then divide by the number of angles 540º ÷ 5 = 108º

FHSUnit G14 Polygon Exterior Angle Sum Theorem The sum of the exterior angle measures of any convex polygon is 360º. Example: Find the sum of the exterior angles of a convex octagon. How would we find one exterior angle of a regular octagon? Divide 360º by the number of sides of the octagon. So 360º ÷ 8 =

FHSUnit G15 How to name a Polygon To name a polygon, list all of the vertices in order in clockwise or counter-clockwise direction. F H G C D B A E Start anywhere and name all vertices in consecutive order: CDEFGHAB or EDCBAHGF are two possible names for this octagon.

FHSUnit G16 Inscribed Quadrilateral Remember from Unit 1: The measure of an inscribed angle is half the measure of its intercepted arc. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. –∠ A and ∠ C are supplementary. –∠ B and ∠ D are supplementary.