1/25 Warm Up Find the value of the indicated measure
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1 1/25 Warm Up Find the value of the indicated measure
2 Lesson 7.1(2 Days) Angles of Polygons
3 Essential Question: What is the sum of the measures of the interior angles of a polygon?
4 What you will learn Use the interior angle measures of polygons. Use the exterior angle measures of polygons.
5 Polygon A plane figure made of three or more segment (sides). Each side intersects exactly two other sides at their endpoints. Polygons are named by vertices in consecutive order, going CW or CCW.
6 Examples Geometry 6.1 Polygons 6
7 Diagonals A segment that joins two non-consecutive vertices. The diagonals from one vertex divide a polygon into triangles.
8 Interior Angle Sum of a Triangle m 1 = m m 2 = m m 4 m 1 + m 5 m 2 + m 3 =
9 Polygon Angle Sums Polygon Sides Triangles Degrees Triangle
10 Angle Sum in a Quadrilateral s 360 m 1 + m 2 + m 3 = m 4 + m 5 + m 6 = 180 m 1 + m 4 + m 2 + m 5 + m 3 + m 6 = 360
11 Polygon Angle Sums Polygon Sides Triangles Degrees Triangle Quadrilateral
12 Angle Sum in a Pentagon From this vertex, how many diagonals are there? 2
13 Angle Sum in a Pentagon How many triangles are there? 3 And what is the sum of the angles of each triangle?
14 Angle Sum in a Pentagon 3 s So what is the sum of the interior angles of a pentagon? = 540
15 Polygon Angle Sums Polygon Sides Triangles Degrees Triangle Quadrilateral Pentagon
16 Angle Sum in a Hexagon How many diagonals from this vertex? 3 How many triangles are formed? 4 The sum of the angles is? = s 720
17 Polygon Angle Sums Polygon Sides Triangles Degrees Triangle Quadrilateral Pentagon Hexagon
18
19 Polygon Angle Sums Polygon Sides Triangles Degrees Triangle Quadrilateral Pentagon Hexagon
20 Draw diagonals from one point.
21 Draw diagonals from one point. 3 sides 1 Triangle 4 sides 2 Triangles 5 sides 3 Triangles 6 sides 4 Triangles
22 8 Sides,? Triangles
23 8 Sides, 6 Triangles
24 Polygon Angle Sums Polygon Sides Triangles Degrees Triangle Quadrilateral Pentagon Hexagon Octagon
25 What s the pattern? A polygon with n sides can be divided into how many triangles? n 2 The sum of the angles then is? 180(n 2)
26 Polygon Angle Sums Polygon Sides Triangles Degrees Triangle Quadrilateral Pentagon Hexagon Octagon n-gon n n 2 180(n 2)
27 The Sum of the Interior Angles No. Sides Sum No. Sides Sum
28 Theorem 7.1 Polygon Interior Angles Theorem The sum of the interior angles of a convex n- gon is: 180 ( n 2) memorize this!
29 Example 1: Find the sum of the measures of the interior angles of the figure. 180 (n 2) = 180 (8 2) = = 1080
30 Example 2: Find the sum of the interior angles of a polygon with 14 sides. 180(14 2) = 180(12) = 2160
31 Example 3: The sum of the interior angles of a polygon is How many sides does the polygon have? (n 2) = n 360 = n = 3060 n = 17 Easier way to solve: 180 n 2 = n 2 = n 2 = n = 17
32 Example 4: The sum of the interior angles of a polygon is How many sides does the polygon have? (n 2) = n 360 = n = 1980 n = 11 Easier way to solve: 180 n 2 = n = n 2 = n = 11
33 Example 5: The sum of the interior angles of a polygon is How many sides does the polygon have? This is not a polygon. 180(n 2) = n 360 = n = 1740 n = This must be a whole number. Easier way to solve: 180 n 2 = n 2 = n 2 = This must be a whole number. January 25, 2016
34 Example 6: Find m E & m F H A B C Solution: The figure is an octagon. The sum of its angles is G D The sum of angles G to D is 840. F E That leaves 240. F E, so each is 120.
35 7.1 Corollary to the Polygon Interior Angles Theorem The sum of the interior angles of a quadrilateral is = 360
36 Example 7: Solve for x. x + x = 360 2x = 360 2x =250 x = 125 x 55 x
37 Your Turn Find the value of x in the diagram. x = 360 x = 360 x = 72
38 Regular Polygons
39 Regular Polygon All sides congruent All angles congruent The Sum of the interior angles is 180 (n 2) Since the angles are congruent, the measure of each angle is 180 ( n 2) n
40 Example 8: Find the measure of each angle of a regular pentagon (5 2) 180(3)
41 Example 9: Find the measure of each angle of a regular octagon (8 2) 180(6)
42 Example 10: Each angle of a regular polygon measures 160. How many sides does the polygon have? 180( n 2) 160 n 180n n 20n 360 n 18
43 Example 11: Each angle of a regular polygon measures 171. How many sides does the polygon have? 180( n 2) 171 n 180n n 9n 360 n 40
44 Example 12: A home plate for a baseball field is shown. a. Is the polygon regular? Explain your reasoning. The polygon is not equilateral or equiangular. So, the polygon is not regular.
45 Example 12: b. Find the measures of C and E. 180 (n 2) = 180 (5 2) = 540 x + x = 540 2x = 540 2x = 270 x = 135 Therefore, C= 135 and E= 135
46 The Sum of the Exterior Angles This always means using one exterior angle at each vertex. But not this one. This angle or this angle.
47 The Sum of the Exterior Angles Extend only ONE side at each vertex. Exterior Angle Exterior Angle Exterior Angle
48 The Sum of the Exterior Angles 180 c? c a?180 a b?180 b
49 The Sum of the Exterior Angles The sum of the exterior angles is: (180 a) + (180 b) + (180 c) 180 c c a 180 a b 180 b
50 The sum of the exterior angles (180 a) + (180 b) + (180 c) a b c 540 (a + b + c) What is (a + b + c)? 180 (this is a triangle!) =? 360
51 What if we had a quadrilateral? There would be four angles: a,b,c,d. The exterior sum would be: (180 a) + (180 b) + (180 c) + (180 d) which is 720 (a + b + c + d) (a + b + c + d) =? = 360 (again!)
52 Theorem 7.2 Polygon Exterior Angles Theorem The sum of the exterior angles of ANY polygon, one angle at each vertex, is 360. m 1 + m m n = 360
53 Example 13: Find the value of x in the diagram. x + 2x = 360 3x = 360 3x = 204 x = 68
54 Corollary The measure of an exterior angle of a regular polygon with n sides is 360 n
55 Example 14: Find the measure of an exterior angle of a regular 40-gon. Solution: 360/40 = 9
56 Example 15: The trampoline shown is shaped like a regular dodecagon. a. Find the measure of each interior angle. 180 ( n 2) n 180(12 2) = 12 = 180(10) 12 = 1800 =
57 Example 15: The trampoline shown is shaped like a regular dodecagon. b. Find the measure of each exterior angle = 30
58 Summary The sum of the interior angles of an n-gon is 180(n 2). The sum of the exterior angles of any polygon is 360. The measure of an interior angle of a regular polygon is 180(n 2) n The measure of an exterior angle of a regular polygon is 360 n..
59 Assignment
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