1. Quadrilaterals Chapter 6

2. Polygons

3. What is a Polygon? • Formed by 3 or more segments (sides) • Each side intersects only 2 other sides (one at each endpoint)

4. What is a Polygon?

5. What’s in a name? Polygons are named by the number of sides they have

6. Classifying Polygons CONVEX CONCAVE

7. Concave or Convex?

8. Classifying Polygons • Regular Polygons: • Equilateral & Equiangular

9. Regular or Irregular?

10. Diagonals of Polygons • Segment that joins 2 non-consecutive vertices.

11. Diagonals

12. Interior Angles of a Quadrilateral Theorem • The Sum of the Measures of the Interior Angles of a Quadrilateral is 360°

13. Interior Angles of Quadrilaterals Solve for x…

14. Parallelograms

15. What is a Parallelogram? • Quadrilateral • Both pairs of opposite sides are parallel

16. Theorems about Parallelograms If a Quadrilateral is a Parallelogram, Then…. • OPPOSITE SIDES are congruent • OPPOSITE ANGLES are congruent

17. Theorems about Parallelograms If a Quadrilateral is a Parallelogram, Then…. • CONSECUTIVE ANGLES are supplementary • DIAGONALS bisect each other A +B= 180°

18. Proving Quadrilaterals are Parallelograms

19. Prove it! Proving Quadrilaterals are Parallelograms… • If both pairs of opposite sides of a quad. are  … • If both pairs of opposite angles of a quad. are  … • If an angle of a quad. is supplementary to both of its consecutive angles … • If the diagonals of a quad. bisect each other… Then, the Quadrilateral is a Parallelogram.

20. Prove it! Proving Quadrilaterals are Parallelograms… • If one pair of opposite sides of a quadrilateral are congruent AND parallel Then, the Quadrilateral is a Parallelogram.

21. Prove it! Let’s practice…. • Describe how to prove that ABCD is a parallelogram given that ∆PBQ  ∆RDS and ∆PAS  ∆RCQ.

22. Prove it! Let’s practice…. • Prove that EFGH is a parallelogram by showing that a pair of opposite sides are both congruent and parallel. • Use E(1, 2), F(7, 9), G(9, 8), and H(3, 1). • Prove that JKLM is a parallelogram by showing that the diagonals bisect each other. • Use J(-4, 4), K(-1, 5), L(1, -1), and M(-2, -2).

23. Quiz 1 Sections 1, 2, & 3

24. Special Parallelograms

25. Rhombus • A parallelogram with 4 congruent sides • Rhombus Corollary: A quadrilateral is a rhombus if and only if it has four congruent sides.

26. Rhombus • Theorem 6.11: • A parallelogram is a rhombus if and only if its diagonals are perpendicular. • ABCD is a rhombus if and only if AC  BD

27. Rhombus • Theorem 6.12: • A parallelogram is a rhombus if and only if its diagonals bisect a pair of opposite angles. • ABCD is a rhombus if and only if AD bisects CAB and BDC and BC bisects DCA and ABD

28. Rectangle • A parallelogram with 4 right angles • Rectangle Corollary: A quadrilateral is a rectangle if and only if it has four right angles.

29. Rectangle • Theorem 6.13: • A parallelogram is a rectangle if and only if its diagonals are congruent. • ABCD is a rectangle if and only if AC  BD

30. Square • A parallelogram with 4 congruent sides AND 4 right angles • Square Corollary: A quadrilateral is a square if and only if it is a rhombus and a rectangle.

31. Special Parallelograms

32. Trapezoids

33. Trapezoids • Quadrilateral with only one pair of parallel sides. • Parallel sides are the “bases” • Non-parallel sides are the “legs” • Has 2 pairs of base angles Base Angles

34. Isosceles Trapezoids • Show that RSTV is a trapezoid…

35. Isosceles Trapezoids • Legs are congruent • If mA = 45°, • What is the measure of B? • What is the measure of C? • What is the measure of D?

36. Isosceles Trapezoids • Theorem 6.14: If a trapezoid is isosceles, then each pair of base angles is congruent • A  D, B  C

37. Isosceles Trapezoids • Theorem 6.15:(Converse to theorem 6.14) If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid • ABCD is an isosceles trapezoid

38. Isosceles Trapezoids • Theorem 6.16: A trapezoid is isosceles if and only if its diagonals are congruent • ABCD is isosceles if and only if AC  BD

39. Trapezoids Midsegment Theorem for Trapezoids (Theorem 6.17) EF  AB, EF  DC, EF = ½(AB + DC) • The midsegment of a trapezoid is … • Parallel to each base • ½ the sum of the length of the bases

40. Kites

41. Kites • A quadrilateral that has two pairs of consecutive congruent sides. • Opposite sides are NOT congruent.

42. Theorems about Kites • Theorem 6.18: If a quadrilateral is a kite, then its diagonals are perpendicular KT  EI

43. Practicing Theorems about Kites • If KS = ST = 5, ES = 4, and KI = 9, • What is the measure of EK? • What is the measure of SI?

44. Theorems about Kites • Theorem 6.19: If a quadrilateral is a kite, then only one pair of opposite angles are congruent K M, J  L

45. Practicing Theorems about Kites • If mJ = 70 and mL = 50, • What is mM & mK?

46. Quiz 2 Sections 4 & 5

47. Special Quadrilaterals

48. Special Quadrilaterals • When you join the midpoints of the sides of ANY quadrilateral, what special quadrilateral is formed? Explain. • On a piece of graph paper… • Draw ANY quadrilateral • Find and connect the midpoints of each side • What type of Quadrilateral is formed? • How do you know?

49. Special Quadrilaterals • Let’s prove a quadrilateral is a “special” shape… • Use the Definition of the Shape • Use a Theorem • EXAMPE: Show that PQRS is a rhombus • How would you prove this to be true?

50. Special Quadrilaterals Create a Graphic Organizer showing the relationship between the following figures… Requirements.. Accurate Graphic Organizers Each figure should include an picture and description Bold, Clear, and Colorful • Isosceles Trapezoid • Kite • Parallelogram • Quadrilaterals • Rectangle • Rhombus • Square • Trapezoid