Download presentation

Presentation is loading. Please wait.

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 π©π« and π©π¬ are diagonals of vertex π©

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 β Polygon Interior Angles Theorem: The sum of the measures of the interior angles of a convex πβπππ is (πβπ)βπππΒ°. Example: π<1+β¦+π<6=(6β2)β180 =4β180=πππ 5 3 2 6 1 4 π=π

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 (8β2)β180 6β180 ππππΒ°

11

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

12

Example 2 Find the number of sides of a convex polygon whose interior angles add to πππΒ°. Give the name of this polygon. πβ2 β180=900

13

Example 2 Find the number of sides of a convex polygon whose interior angles add to πππΒ°. Give the name of this polygon. πβ2 β180=900 πβ2=5 π=π

14

Example 2 Find the number of sides of a convex polygon whose interior angles add to πππΒ°. Give the name of this polygon. The polygon has π=7 sides. Thus, this polygon is a Heptagon πβ2 β180=900 πβ2=5 π=π

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 ππππΒ°. 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 (11β2)β180 9β180 ππππΒ°

17

Example 3 b.) The sum of the measures of the interior angles of a convex polygon is ππππΒ°. Classify the polygon by the number of sides it has. The polygon has π=10 sides. Thus, this polygon is a Decagon. πβ2 β180=1440 πβ2=8 π=ππ

18

Corollary Corollary to Theorem 8.1 β Interior Angles of a Quadrilateral: The sum of the measures of the interior angles of a quadrilateral is πππΒ°. Example: In Quadrilateral ABCD, π<π¨+π<π©+π<πͺ+π<π«=πππΒ° B C A D

19

Example 4 Find the value of x in the diagram shown. πππΒ° πππΒ° πΒ° ππΒ°

20

Example 4 Find the value of x in the diagram shown. πππΒ° πππΒ° πΒ° ππΒ°

π₯ =360Β° π₯+288=360 π=ππΒ°

21

Example 5 Find the value of x in the diagram shown. πππΒ° ππΒ° πππΒ° πΒ°

ππΒ°

22

Example 5 Find the value of x in the diagram shown. πππΒ° ππΒ°

πππΒ° πππΒ° πΒ° ππΒ° ππΒ° The polygon is a Pentagon, with angle measure 5β2 β180=3β180=πππΒ°

23

Example 5 Find the value of x in the diagram shown. πππΒ° ππΒ°

πππΒ° πππΒ° πΒ° ππΒ° ππΒ° π₯ =540 π₯+384=540 π=πππΒ°

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 β Polygon Exterior Angles Theorem: The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is πππΒ°. 5 3 2 1 4 Example: π<1+β¦+π<5=πππΒ° π=π

26

Example 6 Use the polygon exterior angles theorem to solve for the value of x. ππΒ° ππΒ° πΒ° ππΒ°

27

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

28

Example 6 Use the polygon exterior angles theorem to solve for the value of x. ππΒ° ππΒ° πΒ° ππΒ° π₯+2π₯+89+67=360 3π₯+156=360 3π₯=204 π=ππ

29

Example 7 Use the polygon exterior angles theorem to solve for the value of x. (π+ππ)Β° ππΒ° πΒ° πππΒ°

30

Example 7 Use the polygon exterior angles theorem to solve for the value of x. (π+ππ)Β° ππΒ° πΒ° πππΒ° π₯+ π₯ =360

31

Example 7 Use the polygon exterior angles theorem to solve for the value of x. (π+ππ)Β° ππΒ° πΒ° πππΒ° π₯+ π₯ =360 2π₯+228=360 2π₯=132 π=ππ

32

Final Practice 1 The measures of three of the interior angles of a quadrilateral are ππΒ°, πππΒ°, and ππΒ°. Find the measure of the fourth angle.

33

Final Practice 1 The measures of three of the interior angles of a quadrilateral are ππΒ°, πππΒ°, and ππΒ°. Find the measure of the fourth angle. π₯ =360Β° π₯+245=360 π=πππΒ°

34

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

35

Final Practice 2 The sum of the measures of a convex polygon is ππππΒ°. Classify the Polygon by the number of sides it has. The polygon has π=15 sides. Thus, this polygon is a 15-gon. πβ2 β180=2340 πβ2=13 π=ππ

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 (9β2)β180 7β180 ππππΒ°

38

Final Practice 4 A convex hexagon has exterior angles with measures ππΒ°, ππΒ°, ππΒ°, ππΒ°, and ππΒ°. Find the measure of exterior angle at the sixth vertex.

39

Final Practice 4 A convex hexagon has exterior angles with measures ππΒ°, ππΒ°, ππΒ°, ππΒ°, and ππΒ°. Find the measure of exterior angle at the sixth vertex. π₯ =360 π₯+283=360 π=ππΒ°

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 (12β2)β180 10β180 ππππΒ°

42

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

43

Extra Example Find measures of each interior and exterior angle of the regular polygon given. For the Interior Angles: 1800Γ·12=πππΒ° (Why?) For the Exterior Angles: 360Γ·12=ππΒ° (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 πππΒ°, and each exterior angles has a measure of ππΒ°.

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 (18β2)β180 16β180 ππππΒ°

46

Extra Example Find measures of each interior and exterior angle of a regular 18-gon. For the Interior Angles: 2880Γ·18=πππΒ° For the Exterior Angles: 360Γ·18=ππΒ° Thus, each interior angle of a regular 18-gon has a measure of πππΒ°, and each exterior angle has a measure of ππΒ°.

Similar presentations

Β© 2023 SlidePlayer.com Inc.

All rights reserved.