Chapter 6

1. Chapter 6 Quadrilaterals

2. Section 6.1 Polygons

3. Polygon • A polygon is formed by three or more segments called sides • No two sides with a common endpoint are collinear. • Each side intersects exactly two other sides, one at each endpoint. • Each endpoint of a side is a vertex of the polygon. • Polygons are named by listing the vertices consecutively.

4. Identifying polygons • State whether the figure is a polygon. If not, explain why.

5. Polygons are classified by the number of sides they have octagon triangle nonagon quadrilateral pentagon decagon dodecagon hexagon heptagon N-gon

6. Two Types of Polygons: • Convex: If a line was extended from the sides of a polygon, it will NOT go through the interior of the polygon. Example:

7. 2. Concave: If a line was extended from the sides of a polygon, it WILL go through the interior of the polygon. Example:

9. Regular Polygon • A polygon is regular if it is equilateral and equiangular • A polygon is equilateral if all of its sides are congruent • A polygon is equiangular if all of its interior angles are congruent

11. Diagonal • A segment that joins two nonconsecutive vertices.

13. 2 1 3 4 Interior Angles of a Quadrilateral Theorem • The sum of the measures of the interior angles of a quadrilateral is 360°

16. Section 6.2 Properties of Parallelograms

17. Parallelogram • A quadrilateral with both pairs of opposite sides parallel

18. Theorem 6.2 • Opposite sides of a parallelogram are congruent.

19. Theorem 6.3 • Opposite angles of a parallelogramare congruent

20. 2 3 4 1 Theorem 6.4 • Consecutive angles of a parallelogram are supplementary.

21. Theorem 6.5 • Diagonals of a parallelogram bisect each other.

25. Section 6.3 Proving Quadrilaterals are Parallelograms

26. Theorem 6.6To prove a quadrilateral is a parallelogram: • Both pairs of opposite sides are congruent

27. Theorem 6.7 To prove a quadrilateral is a parallelogram: • Both pairs of opposite angles are congruent.

28. 2 3 4 1 Theorem 6.8 To prove a quadrilateral is a parallelogram: • An angle is supplementary to both of its consecutive angles.

29. Theorem 6.9To prove a quadrilateral is a parallelogram: • Diagonals bisect each other.

30. Theorem 6.10 To prove a quadrilateral is a parallelogram: • One pair of opposite sides are congruent and parallel. > >

33. Section 6.4 Rhombuses, Rectangles, and Squares

34. Rhombus • Parallelogram with four congruent sides.

35. Rectangle • Parallelogram with four right angles.

36. Square • Parallelogram with four congruent sides and four congruent angles. • Both a rhombus and rectangle.

37. Theorem 6.11 • Diagonals of a rhombus are perpendicular.

38. Theorem 6.12 • Each Diagonal of a rhombus bisects a pair of opposite angles.

39. Theorem 6.13 • Diagonals of a rectangle are congruent.

40. Section 6.5 Trapezoids and Kites

41. Trapezoid • Quadrilateral with exactly one pair of parallel sides. • Parallel sides are the bases. • Two pairs of base angles. • Nonparallel sides are the legs. Base > Leg Leg > Base

42. Isosceles Trapezoid • Legs of a trapezoid are congruent.

43. > A B > D C Theorem 6.14 • Base angles of an isosceles trapezoid are congruent.

44. Theorem 6.15 • If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid. > A B > D C ABCD is an isosceles trapezoid

45. A B > D C Theorem 6.16 • Diagonals of an isosceles trapezoid are congruent. > ABCD is isosceles if and only if

46. Examples on Board

47. Midsegment of a trapezoid • Segment that connects the midpoints of its legs. Midsegment

48. C B M N A D Midsegment Theoremfor trapezoids • Midsegment is parallel to each base and its length is one half the sum of the lengths of the bases. MN= (AD+BC)

49. Examples on Board

50. Kite • Quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.