Lesson 7.1. Angles of Polygons
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1 Lesson 7.1 Angles of Polygons
2 Essential Question: How can I find the sum of the measures of the interior angles of a polygon?
3 Polygon A plane figure made of three or more segments (sides). Each side intersects exactly two other sides at their endpoints. Polygons are named by vertices in consecutive order, going CW or CCW.
4 Diagonals in a Polygon A segment that joins two non-consecutive vertices. The diagonals from one vertex divide a polygon into triangles.
5 Interior Angle Sum of a Triangle m 1 = m m 2 = m m 4 m 1 + m 5 m 2 + m 3 =
6 Polygon Interior Angle Sums Polygon Sides Triangles formed by Diagonals Sum of Interior Angles Triangle
7 Interiore 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
8 Polygon Interor Angle Sums Polygon Sides Triangles formed by Diagonals Sum of Interior Angles Triangle Quadrilateral
9 Interior Angle Sum in a Pentagon From this vertex, how many diagonals are there? 2
10 Angle Sum in a Pentagon How many triangles are there? 3 And what is the sum of the angles of each triangle?
11 Angle Sum in a Pentagon 3 s So what is the sum of the interior angles of a pentagon? = 540
12 Polygon Interior Angle Sums Polygon Sides Triangles formed by Diagonals Sum of Interior Angles Triangle Quadrilateral Pentagon
13 Interior 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
14 Polygon Interior Angle Sums Polygon Sides Triangles formed by Diagonals Sum of Interior Angles Triangle Quadrilateral Pentagon Hexagon
15
16 Polygon Interior Angle Sums Polygon Sides Triangles formed by Diagonals Sum of Interior Angles Triangle Quadrilateral Pentagon Hexagon Octagon
17 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)
18 Polygon Interior Angle Sums Polygon Sides Triangles formed by Diagonals Sum of Interior Angles Triangle Quadrilateral Pentagon Hexagon Octagon n-gon n n 2 180(n 2)
19 Theorem 7.1 Polygon Interior Angles Theorem The sum of the interior angles of a convex n- gon is: 180 ( n 2) memorize this!
20 Example 1: Find the sum of the interior angles of a polygon with 14 sides. 180(14 2) = 180(12) = 2160
21 Example 2: 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
22 Example 3: 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
23 Example 4: 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 = Why must this be a whole number?
24 7.1 Corollary to the Polygon Interior Angles Theorem The sum of the interior angles of a quadrilateral is = 360
25 Example 5: Solve for x. x 55 x x + x = 360 2x = 360 2x = 250 x = 125
26 Your Turn Find the value of x in the diagram. x = 360 x = 360 x = 72
27 Regular Polygons
28 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 interior angle in a regular polygon is E I = 180 (n 2) n
29 Example 6: Find the measure of each angle of a regular pentagon (5 2) 180(3)
30 Example 7: 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
31 Example 8: 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.
32 Example 8: 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
33 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.
34 The Sum of the Exterior Angles Extend only ONE side at each vertex. Exterior Angle Exterior Angle Exterior Angle
35 Theorem 7.2 Polygon Exterior Angles Theorem The sum of the exterior angles of any polygon, one angle at each vertex, is 360. S E = 360 m 1 + m m n = 360
36 Example 9: Find the value of x in the diagram. x + 2x = 360 3x = 360 3x = 204 x = 68
37 Corollary The measure of an exterior angle of a regular polygon with n sides is E = 360 E n
38 Example 10: Find the measure of an exterior angle of a regular 40-gon. Solution: 360/40 = 9
39 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 =
40 Example 15: The trampoline shown is shaped like a regular dodecagon. b. Find the measure of each exterior angle = 30
41 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..
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