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
Review 4 5 3
m2 = m5
m1 = m4
180 1 2
m1 + m2 + m3 = 180
m4 + m5 + m3 = 180
1   180
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6. Polygon Interior Angle Sums
Sides
Triangles formed by Diagonals
Sum of Interior Angles
Triangle 3 1
180
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7. Interiore Angle Sum in a Quadrilateral2 3 6 1 4
m1 + m2 + m3 = 180
m4 + m5 + m6 = 180 5
m1 + m4 + m2 + m5 + m3 + m6 = 360
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8. Polygon Interor Angle Sums
Sides
Triangles formed by Diagonals
Sum of Interior Angles
Triangle 3 1
180
Quadrilateral 4 2
360
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9. Interior Angle Sum in a Pentagon
From this vertex, how many diagonals are there? 2
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10. Angle Sum in a Pentagon 3 180 180 180 180
How many triangles are there?
180 3
180
And what is the sum of the angles of each triangle?
180
180
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11. Angle Sum in a Pentagon 3  180 = 540 3 s  540 180 180 180
So what is the sum of the interior angles of a pentagon?
180
180
3  180 = 540
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12. Polygon Interior Angle Sums
Sides
Triangles formed by Diagonals
Sum of Interior Angles
Triangle 3 1
180
Quadrilateral 4 2
360
Pentagon 5
540
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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?
4  180 = 720
4 s  720
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14. Polygon Interior Angle Sums
Sides
Triangles formed by Diagonals
Sum of Interior Angles
Triangle 3 1
180
Quadrilateral 4 2
360
Pentagon 5
540
Hexagon 6
720
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15. Do you see a pattern? September 18, 2018September 18, 2018
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16. Polygon Interior Angle Sums
Sides
Triangles formed by Diagonals
Sum of Interior Angles
Triangle 3 1
180
Quadrilateral 4 2
360
Pentagon 5
540
Hexagon 6
720
Octagon 8
1080
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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)
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18. Polygon Interior Angle Sums
Sides
Triangles formed by Diagonals
Sum of Interior Angles
Triangle 3 1
180
Quadrilateral 4 2
360
Pentagon 5
540
Hexagon 6
720
Octagon 8
1080
n-gon n
n – 2
180(n – 2)
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19. Theorem 7.1 Polygon Interior Angles Theorem
The sum of the interior angles of a convex n-gon is:
memorize this!
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20. Example 1:
Find the sum of the interior angles of a polygon with 14 sides.
180(14 – 2) = 180(12)
= 2160
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21. Example 2:
The sum of the interior angles of a polygon is 2700. How many sides does the polygon have? 17
180(n – 2) = 2700
180n – 360 = 2700
180n = 3060
n = 17
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22. Example 3:
The sum of the interior angles of a polygon is 1620. How many sides does the polygon have? 11
180(n – 2) = 1620
180n – 360 = 1620
180n = 1980
n = 11
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23. Example 4:
The sum of the interior angles of a polygon is 1380. How many sides does the polygon have?
This is not a polygon.
180(n – 2) = 1380
180n – 360 = 1380
180n = 1740
n = …
Why must this be a whole number?
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24. 7.1 Corollary to the Polygon Interior Angles Theorem
The sum of the interior angles of a quadrilateral is 360.

25. Example 5: x + x + 55 + 55 = 360 2x + 110 = 360 2x = 250 x = 125 °
Solve for x.
x + x = 360
2x = 360
2x = 250
x = 125 °

26. Your Turn
Find the value of x in the diagram. x° + 108° + 121° + 59° = 360° x = 360 x = 72 °

27. Regular Polygons

28. Regular Polygon 𝐸 𝐼 = 180°(𝑛 −2) 𝑛 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
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7.1 Angles of Polygons

29. Example 6: Find the measure of each angle of a regular pentagon. 108
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30. Example 7:
Each angle of a regular polygon measures 160. How many sides does the polygon have?
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7.1 Angles of Polygons

31. Example 8: A home plate for a baseball field is shown.
Is the polygon regular? Explain your reasoning.
The polygon is not equilateral or equiangular. So, the polygon is not regular.

32. Example 8: Find the measures of ∠C and ∠E. 180° ⋅ (n − 2)
x° + x° + 90° + 90° + 90° = 540°
2x = 540
2x = 270
x = 135
Therefore, ∠C= 135° and ∠E= 135°
= 180° ⋅ (5 − 2)
= 540°

33. The Sum of the Exterior Angles
This always means using one exterior angle at each vertex.
But not this one.
or this angle.
This angle…
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34. The Sum of the Exterior Angles
Extend only ONE side at each vertex.
Exterior Angle
Exterior Angle
Exterior Angle
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35. 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∠2 + · · · + m∠n = 360°
𝑆 𝐸 =360°

36. Example 9:
Find the value of x in the diagram. x° + 2x° + 89° + 67° = 360° 3x = 360 3x = 204 x = 68 °

37. Corollary
The measure of an exterior angle of a regular polygon with n sides is
𝐸 𝐸 = 360° 𝑛
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38. Example 10: Find the measure of an exterior angle of a regular 40-gon.
Solution:
360/40 = 9
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39. Example 15: The trampoline shown is shaped like a regular dodecagon.
Find the measure of each interior angle.
= 180(12−2) 12
= 180(10) 12 =
=150°

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 (𝑛−2) 𝑛 .
The measure of an exterior angle of a regular polygon is 𝑛 .
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