Copyright © 2014 Pearson Education, Inc. 3 CHAPTER 3.3 Polygons and Angles Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Definitions Polygon Definition A figure is a polygon if it meets the following conditions: 1. It is a plane figure formed by three or more line segments called sides. 2. Sides that have a common endpoint are noncollinear. 3. Each side intersects exactly two other sides, but only at their endpoints. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Polygons The endpoints of the sides of a polygon are called the vertices (singular is vertex). Below are some examples of polygons. Each vertex of the middle polygon is labeled. A polygon can be named by listing its vertices consecutively in order, as shown. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Names of Polygons Number of Sides Name of Polygon 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10 decagon 12 dodecagon n n-gon Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Identifying Polygons Identify the polygons. If not a polygon, state why. a. b. c. d. e. Solution Figures A and E are polygons. Figure B is not a polygon because there is a “side” that is a curve and not a line segment. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Identifying Polygons Identify the polygons. If not a polygon, state why. a. b. c. d. e. Solution Figure C is not a polygon by our definition because there is a side that intersects more than two other sides. Figure D is not a polygon because two sides intersect only one other side. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Definitions In general, a polygon with n sides is called an n-gon. For example, a polygon with 13 sides is called a 13-gon. Another way to classify polygons is as convex or concave. A polygon is convex if no line containing a side contains a point within the interior of the polygon. A polygon is concave (or nonconvex) if it is not convex. Copyright © 2014 Pearson Education, Inc.

Identifying Convex and Concave Polygons Identify the polygons. If not a polygon, state why. a. b. c. Solution a. The polygon has 8 sides, so it is an octagon. None of the extended sides contain a point of the interior, so it is convex. Copyright © 2014 Pearson Education, Inc.

Identifying Convex and Concave Polygons Identify the polygons. If not a polygon, state why. a. b. c. Solution b. The polygon has 6 sides, so it is a hexagon. Some of the extended sides contain a point of the interior, so it is concave (or nonconvex). Copyright © 2014 Pearson Education, Inc.

Identifying Convex and Concave Polygons Identify the polygons. If not a polygon, state why. a. b. c. Solution c. The polygon has 4 sides, so it is a quadrilateral. None of the extended sides contain a point within the interior, so it is convex. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Definition An equilateral polygon is a polygon with all sides congruent. An equiangular polygon is a polygon with all angles congruent. A regular polygon is a polygon that is both equilateral and equiangular. Copyright © 2014 Pearson Education, Inc.

Identifying Regular Polygons Determine if each polygon is regular or not. Explain your reasoning. a. b. c. Solution a. The pentagon is equilateral and equiangular, so it is a regular polygon. Copyright © 2014 Pearson Education, Inc.

Identifying Regular Polygons Determine if each polygon is regular or not. Explain your reasoning. a. b. c. Solution b. The hexagon is equilateral, but not equiangular, so it is not a regular polygon. Copyright © 2014 Pearson Education, Inc.

Identifying Regular Polygons Determine if each polygon is regular or not. Explain your reasoning. a. b. c. Solution c. The quadrilateral is equilateral, but not equiangular, so it is not a regular polygon. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Definitions The perimeter P of a polygon is the sum of the lengths of its sides. Copyright © 2014 Pearson Education, Inc.

Finding the Perimeter of an Irregular Room Find the perimeter of the room. Solution To find the perimeter of the room, we first need to find the lengths of all sides of the room. Copyright © 2014 Pearson Education, Inc.

Finding the Perimeter of an Irregular Room Add the measures to find the perimeter. perimeter = 10 ft + 9 ft + 3 ft + 6 ft + 7 ft + 15 ft = 50 ft The perimeter of the room is 50 feet. Copyright © 2014 Pearson Education, Inc.

Triangle Interior Angle Sum Worksheet   Copyright © 2014 Pearson Education, Inc.

Corollary 3.14 Exterior Angle of a Triangle The measure of each exterior angle of a triangle equals the sum of the measures of its two nonadjacent interior angles. m∠1 = m∠2 + m∠3 Copyright © 2014 Pearson Education, Inc.

Corollary 3.13 Third Angles Theorem Copyright © 2014 Pearson Education, Inc.

Using the Third Angles Theorem Find the value of x. Solution From the figures, we have ∠J ≅ ∠R and ∠H ≅ ∠S. Thus, from the Third Angles Theorem, ∠K ≅ ∠Q or m∠K = m∠Q. Use ΔJHK to find m∠K. m∠K = 180° – 53° – 92° = 35° Copyright © 2014 Pearson Education, Inc.

Using the Third Angles Theorem Find the value of x. Solution m∠K = m∠Q 35 = 10x + 5 30 = 10x 3 = x The value of x is 3. To check, replace x with 3 and see that 10x + 5 = 35. Then make sure that 53° + 92° + 35° = 180°. Copyright © 2014 Pearson Education, Inc.

Finding Angle Measures Use the Triangle Angle-Sum Theorem to find the measure of each angle in the given triangle. Solution 5x + 6x + 15x + 24 = 180 26x + 24 = 180 26x = 156 x = 6 Now let’s use the value of x and the given triangle to find the measure of each angle. Copyright © 2014 Pearson Education, Inc.

Finding Angle Measures Use the Triangle Angle-Sum Theorem to find the measure of each angle in the given triangle. Solution If x = 6, then 5x = 5(6) = 30 6x = 6(6) = 36 15x + 24 = 15(6) + 24 = 90 + 24 = 114 Check: 30° + 36° + 114° = 180° Copyright © 2014 Pearson Education, Inc.

Finding Angle Measures Use the Exterior Angle of a Triangle Corollary to find the measure of the exterior angle and the nonadjacent angle shown. Solution 3x – 53 = x + 67 2x – 3 = 67 2x = 120 x = 60 Copyright © 2014 Pearson Education, Inc.

Finding Angle Measures Solution Let’s use the value of x and the given figure to find the measure of each angle. x = 60 3x – 53 = 3(60) – 53 = 180 – 53 = 127 Also, x° = 60° Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Definition A segment joining two nonconsecutive vertices of a convex polygon is called a diagonal of the polygon. Copyright © 2014 Pearson Education, Inc.

Interior Angle Sum Worksheet Copyright © 2014 Pearson Education, Inc.

Theorem 3.15 Polygon Interior Angle-Sum Theorem The sum of the measures of the interior angles of a convex n-gon is (n – 2) * 180° Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Corollary 3.16 Regular Polygon Interior Angle Corollary (Theorem to 10.1-1) The measure of each interior angle of a regular n-gon is Copyright © 2014 Pearson Education, Inc.

Finding the Sum of the Measures of the Angles of a Polygon Find the sum of the measures of the interior angles of a convex octagon. Solution An octagon has 8 sides. The interior angle sum of a convex octagon is 1080°. Copyright © 2014 Pearson Education, Inc.

Finding the Measure of an Interior Angle Find the value of x in the figure. Then use x to find Solution This is a hexagon, which has 6 sides. The sum of the interior angles of any convex hexagon is: sum of angles = (6 – 2)  180° = 4  180° = 720° Copyright © 2014 Pearson Education, Inc.

Finding the Measure of an Interior Angle To find x, let’s add the interior angle measures of the polygon and set the sum equal to 720°. Copyright © 2014 Pearson Education, Inc.

Using the Regular Polygon Interior-Angle Corollary The Sino-Steel Tower is a hexagonal, honey comb-looking “green” building, in Tianjin, China, designed by MAD Studios architects. Find the measure of each interior angle of one regular hexagon. Solution Copyright © 2014 Pearson Education, Inc.

Finding the Number of Sides of a Regular Polygon The measure of an interior angle of a regular polygon is 144°. Find the number of sides of this polygon. Solution The regular polygon has 10 sides. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Solve for x Find x, and then the measures of angles C and D. Solution x + (2x + 3) + 127 + 125 = 360 3x + 255 = 360 3x = 105 x = 35 Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Solve for x Find x, and then the measures of angles C and D. Solution Use x = 35 to find m∠C and m∠D. m∠D = x° = 35° m∠C = (2x + 3)° = 2(35) + 32° = 73° Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Exterior Angles The angles that are adjacent to the interior angles of a convex polygon are the exterior angles of the polygon. Exterior angles are 4, 5, 6, 7, 8 and 9 Copyright © 2014 Pearson Education, Inc.

Theorem 3.17 Polygon Exterior Angle-Sum Theorem Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Corollary 3.18 Regular Polygon Exterior Angle Corollary (to Theorem 3.17) Copyright © 2014 Pearson Education, Inc.

Finding the Number of Sides of a Regular Polygon Find the measure of each exterior angle of a regular pentagon. Solution Since this is a regular polygon, each exterior angle has the same measure. Each exterior angle measures 72 degrees. Copyright © 2014 Pearson Education, Inc.

Finding the Number of Sides of a Regular Polygon Find the value of x. Then find each exterior angle measure. Solution The sum of the exterior angles is 360° Copyright © 2014 Pearson Education, Inc.

Finding the Number of Sides of a Regular Polygon The angle measures are: Copyright © 2014 Pearson Education, Inc.