Warm-Up

1. Warm-Up • 1) Find the value of 3a2 – 5(a - 4) when a = 2. • 2) Simplify. 12x4 + 3x2y2 – 2y4 + 9y4 -15x4 + 2x4 – 8x2y2 • 3) Simplify. (3m – 4n)(7m + 8n) • 4) Find the area of a triangle whose altitude is 26 m and base is 19 m. 22 -x4 – 5x2y2 + 7y4 21m2 – 4mn + 32m2 A = 247

2. Topics Covered So Far

3. Properties of Polygons

4. What Do You Know?

5. Definitions: SIDE • Polygon—a plane figure that meets the following conditions: • It is formed by 3 or more segments called sides, such that no two sides with a common endpoint are collinear. • Each side intersects exactly two other sides, one at each endpoint. • Vertex – each endpoint of a side. You can name a polygon by listing its vertices consecutively. • Example: PQRST and QPTSR are two correct names for the polygon above.

6. Example 1: Identifying Polygons • State whether the figure is a polygon. If it is not, explain why. • Not D – has a side that isn’t a segment – it’s an arc. • Not E– because two of the sides intersect only one other side. • Not F because some of its sides intersect more than two sides Figures A, B, and C are polygons.

7. Polygons are named by the number of sides they have – Memorize!

8. Convex or concave? • Convex polygon—polygon where the lines containing each side do not contains a point in the interior of the polygon. • Concave polygon –polygon where the lines containing each side do contain a point on the interior of the polygon.

9. Convex or concave? • Identify the polygon and state whether it is convex or concave.

10. Classifying Polygons • Equilateral—all sides of the polygon are congruent • Equiangular—all of the polygon’s interior angles are congruent • Regular—polygon is equilateral and equiangular

11. Identifying Regular Polygons • Decide whether the following polygons are regular.

12. Types of Triangles • Classified by Sides • Equilateral • Isosceles • Scalene • Classified by Angles • Equiangular • Acute • Right • Obtuse

13. Parts of a Triangle • Vertex • Adjacent Sides • Base • Legs • Hypotenuse

14. Additional Triangle Parts • Interior Angle • Exterior Angle 3 2 1

15. Example 1) Find measure of ∠1 2) Find value of x. 1 35° 65° 42° 113° x

16. Interior angles of quadrilaterals • A diagonal of a polygon is a segment that joins two nonconsecutive vertices. • Example: Polygon PQRST has 2 diagonals from point Q, QT and QS diagonals

17. Interior angles of quadrilaterals • Draw a quadrilateral. • Draw a diagonal through it. • How many triangles do you see? • How many degrees do you think are inside a quadrilateral?

18. Theorem 6.1: Interior Angles of a Quadrilateral • The sum of the measures of the interior angles of a quadrilateral is 360°. m1 + m2 + m3 + m4 = 360°

19. Ex. 4: Interior Angles of a Quadrilateral • Find mQ and mR. 80° 70° 2x° x°

20. Practice • Textbook p. 198-199 #16-26 e, 32-38 e • P. 325-326 #4-30, 41-43

22. Homework • Worksheet

23. 11.1 Angle Measures in Polygons Geometry

24. Warm-Up: • Draw each shape; draw a diagonal through each shape; copy and complete this table

25. Homework ?s

26. Polygon Interior Angles Theorem • The sum of the measures of the interior angles of a convex n-gon is (n – 2) ● 180 • The measure of each interior angle of a regular n-gon is: ● (n-2) ● 180 or

27. Ex. 1: Finding measures of Interior Angles of Polygons • Find the value of x in the diagram shown: 142 88 136 105 136 x

28. Ex. 2: Finding the Number of Sides of a Polygon • The measure of each interior angle is 140. How many sides does the polygon have?

29. Polygon Exterior Angles Theorem • The sum of the exterior angles of a convex polygon, with one angle at each vertex, is 360 degrees • The measure of each exterior angle of a regular n-gon is ● 360 or

30. Ex. 3: Finding the Measure of an Exterior Angle

31. Using Angle Measures in Real LifeEx. 5: Using Angle Measures of a Regular Polygon Sports Equipment: If you were designing the home plate marker for some new type of ball game, would it be possible to make a home plate marker that is a regular polygon with each interior angle having a measure of: • 135°? • 145°?

32. Practice • Find the measure of each exterior angle for each regular polygon. Find the sum of the polygon’s interior angles. • 1) Dodecagon 2) 11-sided polygon • 3) 21-gon 4) 15-gon

33. Review of Material • Always, sometimes, never. • 1) An isosceles triangle is __________ an equilateral triangle. • 2) An obtuse triangle is __________ an isosceles triangle. • 3) A triangle __________ has a right angle and an obtuse angle. • 4) An interior angle of a triangle and one of its exterior angles are __________ supplementary.

34. Additional Practice • Textbook p. 665 #14-19, 22, 24 • P. 198-199 #16-20 e, 32-38 e • P. 325-326 #4-30, 41-43

36. Homework

37. Warm-Up • Find the measures of angles 1, 2, and 3 • Are the triangles congruent? Why? 95° 33° 40° 3 1 66° 2

38. ?s

39. Quiz

40. Review • Polygon Formulas • Worksheet

41. Warm-Up • 1) Given a nonagon, • A) Find the sum of the measures of its interior angles. • B) If it were regular, what would the measure of each interior angle be? • C) Find the sum of the measures of its exterior angles. • D) If it were regular, what would the measure of each exterior angle be?

42. ?s

43. Isosceles Triangles • Base • Legs • Base angles • Vertex angle

44. Base Angles Theorem • If two sides of a triangle are congruent, then the angles opposite those sides are congruent

45. Equilateral Triangles • If a triangle is equilateral, then it is equiangular

46. Quick Note • How many degrees are inside a triangle? • If a triangle is equiangular, what are the measures of each angle?

47. Using the Theorems • Find values of x and y. y° 63° x°

48. Using the Theorems • Find values of x and y. x° y°

49. Example 50° y° x°

50. Example y° x° 50°