1. 8-1: Find angle measures in polygons
Geometry Chapter 8
8-1: Find angle measures in polygons

2. Warm-Up
List all of the different polygons we have discussed in this class.
Reminder, the names are determined by the number of sides the polygon has.

3. Find Angle Measures in Polygons
Objective: Students will be able to find angle measures in various polygons using their structures.
Agenda
Definitions
Theorems
Practice

4. Diagonals
A diagonal of a polygon is a segment that joins two nonconsecutive vertices.
Example: B C E A D
Diagonals
𝑩𝑫 and 𝑩𝑬 are diagonals of vertex 𝑩

5. Diagonals
A diagonal of a polygon is a segment that joins two nonconsecutive vertices.
Knowledge Connection:
How many triangles are made by the diagonals? B C E A D

6. Diagonals
A diagonal of a polygon is a segment that joins two nonconsecutive vertices.
Knowledge Connection:
How many triangles are made by the diagonals?
What is the combined measure of all the triangles? B C E A D

7. Theorem 8.1
Theorem 8.1 – Polygon Interior Angles Theorem: The sum of the measures of the interior angles of a convex 𝑛−𝑔𝑜𝑛 is (𝒏−𝟐)∙𝟏𝟖𝟎°.
Example:
𝑚<1+…+𝑚<6=(6−2)∙180
=4∙180=𝟕𝟐𝟎 5 3 2 6 1 4
𝒏=𝟔

8. Example 1
Find the sum of the measures of the interior angles of the convex polygon given.

9. Example 1 The given polygon has 8 sides (its an Octagon).
Find the sum of the measures of the interior angles of the convex polygon given.
The given polygon has 8 sides (its an Octagon).

10. Example 1 The given polygon has 8 sides (its an Octagon).
Find the sum of the measures of the interior angles of the convex polygon given.
The given polygon has 8 sides (its an Octagon).
Thus, its measure is
(8−2)∙180
6∙180
𝟏𝟎𝟖𝟎°

11. Example 2
Find the number of sides of a convex polygon whose interior angles add to 𝟗𝟎𝟎°. Give the name of this polygon.

12. Example 2
Find the number of sides of a convex polygon whose interior angles add to 𝟗𝟎𝟎°. Give the name of this polygon.
𝑛−2 ∙180=900

13. Example 2
Find the number of sides of a convex polygon whose interior angles add to 𝟗𝟎𝟎°. Give the name of this polygon.
𝑛−2 ∙180=900
𝑛−2=5
𝒏=𝟕

14. Example 2
Find the number of sides of a convex polygon whose interior angles add to 𝟗𝟎𝟎°. Give the name of this polygon.
The polygon has 𝑛=7 sides.
Thus, this polygon is a Heptagon
𝑛−2 ∙180=900
𝑛−2=5
𝒏=𝟕

15. Example 3
a.) A coin has the shape of a regular 11-gon. Find the sum of the measures of its interior angles.
b.) The sum of the measures of the interior angles of a convex polygon is 𝟏𝟒𝟒𝟎°. Classify the polygon by the number of sides it has.

16. Example 3
a.) A coin has the shape of a regular 11-gon. Find the sum of the measures of its interior angles.
An 11-gon has 11 sides
Thus, its measure is
(11−2)∙180
9∙180
𝟏𝟔𝟐𝟎°

17. Example 3
b.) The sum of the measures of the interior angles of a convex polygon is 𝟏𝟒𝟒𝟎°. Classify the polygon by the number of sides it has.
The polygon has 𝑛=10 sides.
Thus, this polygon is a Decagon.
𝑛−2 ∙180=1440
𝑛−2=8
𝒏=𝟏𝟎

18. Corollary
Corollary to Theorem 8.1 – Interior Angles of a Quadrilateral: The sum of the measures of the interior angles of a quadrilateral is 𝟑𝟔𝟎°.
Example:
In Quadrilateral ABCD,
𝒎<𝑨+𝒎<𝑩+𝒎<𝑪+𝒎<𝑫=𝟑𝟔𝟎° B C A D

19. Example 4
Find the value of x in the diagram shown.
𝟏𝟎𝟖°
𝟏𝟐𝟏° 𝒙°
𝟓𝟗°

20. Example 4 Find the value of x in the diagram shown. 𝟏𝟎𝟖° 𝟏𝟐𝟏° 𝒙° 𝟓𝟗°
𝑥 =360°
𝑥+288=360
𝒙=𝟕𝟐°

21. Example 5 Find the value of x in the diagram shown. 𝟏𝟏𝟎° 𝟗𝟐° 𝟏𝟎𝟎° 𝒙°
𝟖𝟒°

22. Example 5 Find the value of x in the diagram shown. 𝟏𝟏𝟎° 𝟗𝟐°
𝟏𝟎𝟎°
𝟏𝟏𝟎° 𝒙°
𝟗𝟐°
𝟖𝟒°
The polygon is a Pentagon, with angle measure
5−2 ∙180=3∙180=𝟓𝟒𝟎°

23. Example 5 Find the value of x in the diagram shown. 𝟏𝟏𝟎° 𝟗𝟐°
𝟏𝟎𝟎°
𝟏𝟏𝟎° 𝒙°
𝟗𝟐°
𝟖𝟒°
𝑥 =540
𝑥+384=540
𝒙=𝟏𝟓𝟔°

24. Exterior Angles
The sum of the measures of the Interior Angles depends on the number of sides the polygon has.
Exterior Angles, however, always have the same angle measure, regardless of the number of sides the polygon has.

25. Theorem 8.2
Theorem 8.2 – Polygon Exterior Angles Theorem: The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 𝟑𝟔𝟎°. 5 3 2 1 4
Example:
𝑚<1+…+𝑚<5=𝟑𝟔𝟎°
𝒏=𝟓

26. Example 6
Use the polygon exterior angles theorem to solve for the value of x.
𝟐𝒙°
𝟖𝟗° 𝒙°
𝟔𝟕°

27. Example 6
Use the polygon exterior angles theorem to solve for the value of x.
𝟐𝒙°
𝟖𝟗° 𝒙°
𝟔𝟕°
𝑥+2𝑥+89+67=360

28. Example 6
Use the polygon exterior angles theorem to solve for the value of x.
𝟐𝒙°
𝟖𝟗° 𝒙°
𝟔𝟕°
𝑥+2𝑥+89+67=360
3𝑥+156=360
3𝑥=204
𝒙=𝟔𝟖

29. Example 7
Use the polygon exterior angles theorem to solve for the value of x.
(𝒙+𝟐𝟎)°
𝟗𝟔° 𝒙°
𝟏𝟏𝟐°

30. Example 7
Use the polygon exterior angles theorem to solve for the value of x.
(𝒙+𝟐𝟎)°
𝟗𝟔° 𝒙°
𝟏𝟏𝟐°
𝑥+ 𝑥 =360

31. Example 7
Use the polygon exterior angles theorem to solve for the value of x.
(𝒙+𝟐𝟎)°
𝟗𝟔° 𝒙°
𝟏𝟏𝟐°
𝑥+ 𝑥 =360
2𝑥+228=360
2𝑥=132
𝒙=𝟔𝟔

32. Final Practice 1
The measures of three of the interior angles of a quadrilateral are 𝟖𝟗°, 𝟏𝟏𝟎°, and 𝟒𝟔°. Find the measure of the fourth angle.

33. Final Practice 1
The measures of three of the interior angles of a quadrilateral are 𝟖𝟗°, 𝟏𝟏𝟎°, and 𝟒𝟔°. Find the measure of the fourth angle.
𝑥 =360°
𝑥+245=360
𝒙=𝟏𝟏𝟓°

34. Final Practice 2
The sum of the measures of a convex polygon is 𝟐𝟑𝟒𝟎°. Classify the Polygon by the number of sides it has.

35. Final Practice 2
The sum of the measures of a convex polygon is 𝟐𝟑𝟒𝟎°. Classify the Polygon by the number of sides it has.
The polygon has 𝑛=15 sides.
Thus, this polygon is a 15-gon.
𝑛−2 ∙180=2340
𝑛−2=13
𝒏=𝟏𝟓

36. Final Practice 3
Find the sum of the measures of the interior angles of a convex nonagon.

37. Final Practice 3
Find the sum of the measures of the interior angles of a convex nonagon.
A nonagon has 9 sides
Thus, its measure is
(9−2)∙180
7∙180
𝟏𝟐𝟔𝟎°

38. Final Practice 4
A convex hexagon has exterior angles with measures 𝟑𝟒°, 𝟒𝟗°, 𝟓𝟖°, 𝟔𝟕°, and 𝟕𝟓°. Find the measure of exterior angle at the sixth vertex.

39. Final Practice 4
A convex hexagon has exterior angles with measures 𝟑𝟒°, 𝟒𝟗°, 𝟓𝟖°, 𝟔𝟕°, and 𝟕𝟓°. Find the measure of exterior angle at the sixth vertex.
𝑥 =360
𝑥+283=360
𝒙=𝟕𝟕°

40. Extra Example
Find measures of each interior and exterior angle of the regular polygon given.

41. Extra Example
Find measures of each interior and exterior angle of the regular polygon given.
The given polygon has 12 sides (its a Dodecagon).
Thus, its measure is
(12−2)∙180
10∙180
𝟏𝟖𝟎𝟎°

42. Extra Example
Find measures of each interior and exterior angle of the regular polygon given.
For the Interior Angles:
1800÷12=𝟏𝟓𝟎° (Why?)

43. Extra Example
Find measures of each interior and exterior angle of the regular polygon given.
For the Interior Angles:
1800÷12=𝟏𝟓𝟎° (Why?)
For the Exterior Angles:
360÷12=𝟑𝟎° (Why?)

44. Extra Example
Find measures of each interior and exterior angle of the regular polygon given.
Thus, each interior angle of a regular dodecagon has a measure of 𝟏𝟓𝟎°, and each exterior angles has a measure of 𝟑𝟎°.

45. Extra Example
Find measures of each interior and exterior angle of a regular 18-gon.
An 18-gon has 18 sides
Thus, its measure is
(18−2)∙180
16∙180
𝟐𝟖𝟖𝟎°

46. Extra Example
Find measures of each interior and exterior angle of a regular 18-gon.
For the Interior Angles:
2880÷18=𝟏𝟔𝟎°
For the Exterior Angles:
360÷18=𝟐𝟎°
Thus, each interior angle of a regular 18-gon has a measure of 𝟏𝟔𝟎°, and each exterior angle has a measure of 𝟐𝟎°.