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CHAPTER
3.3 Polygons and Angles
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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.
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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.
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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
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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.
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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.
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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.
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8. 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.
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9. 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).
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10. 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.
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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.
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12. 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.
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13. 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.
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14. 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.
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Definitions
The perimeter P of a polygon is the sum of the lengths of its sides.
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16. 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.
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17. 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.
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18. Triangle Interior Angle Sum Worksheet
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19. 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
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20. Corollary 3.13 Third Angles Theorem
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21. 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°
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22. Using the Third Angles Theorem
Find the value of x. Solution m∠K = m∠Q 35 = 10x = 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°.
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23. 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 = x + 24 = x = 156 x = 6 Now let’s use the value of x and the given triangle to find the measure of each angle.
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24. 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 = = 114 Check: 30° + 36° + 114° = 180°
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25. 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 x – 3 = 67 2x = 120 x = 60
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26. 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°
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Definition
A segment joining two nonconsecutive vertices of a convex polygon is called a diagonal of the polygon.
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28. Interior Angle Sum Worksheet
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29. 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°
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Corollary 3.16 Regular Polygon Interior Angle Corollary (Theorem to )
The measure of each interior angle of a regular n-gon is
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31. 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°.
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32. 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°
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33. 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°.
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34. 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
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35. 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.
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Solve for x
Find x, and then the measures of angles C and D. Solution x + (2x + 3) = 360 3x = 360 3x = 105 x = 35
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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°
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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
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39. Theorem 3.17 Polygon Exterior Angle-Sum Theorem
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Corollary 3.18 Regular Polygon Exterior Angle Corollary (to Theorem 3.17)
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41. 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.
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42. 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°
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43. Finding the Number of Sides of a Regular Polygon
The angle measures are:
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