2. Q UADRILATERALS O BJECTIVES : D EFINE AND CLASSIFY QUADRILATERALS ALONG WITH THEIR RELATED PARTS (P ARALLELOGRAM, R HOMBUS, R ECTANGLE, S QUARE, T RAPEZOID, K ITE ) Homework: Read pg.64-65 pg.66 # 7-10, 13(!)

3. O BJECTIVES To identify any quadrilateral, by name, as specifically as you can, based on its characteristics

4. Q UADRILATERAL a quadrilateral is a polygon with 4 sides.

5. S PECIFIC Q UADRILATERALS There are several specific types of quadrilaterals. They are classified based on their sides or angles.

7. A quadrilateral simply has 4 sides – no other special requirements.

8. E XAMPLES OF Q UADRILATERALS

9. A parallelogram has two pairs of parallel sides.

10. P ARALLELOGRAM Two pairs of parallel sides opposite sides are actually congruent.

11. A rhombus is a parallelogram that has four congruent sides.

12. R HOMBUS Still has two pairs of parallel sides; with opposite sides congruent. 4 in.

13. A rectangle has four right angles.

14. R ECTANGLE Still has two pairs of parallel sides; with opposite sides congruent. Has four right angles

15. A square is a specific case of both a rhombus AND a rectangle, having four right angles and 4 congruent sides.

16. S QUARE Still has two pairs of parallel sides. Has four congruent sides Has four right angles

17. A trapezoid has only one pair of parallel sides.

18. An isosceles trapezoid is a trapeziod with the non-parallel sides congruent.

19. T RAPEZOID has one pair of parallel sides. Isosceles trapezoid trapezoids (Each of these examples shown has top and bottom sides parallel.)

20. An kite is a quadrilateral with NO parallel sides but 2 pairs of adjacent congruent sides.

21. E XAMPLE OF A K ITE 2 in. 4 in. 2 in.

24. C ARNEGIE Homework: Read pg.64-65 pg.66 # 7-10, 13(!)

25. T RAPEZOIDS AND K ITES Chapter 5.3 Homework: pg. 272 # 5, 6, 8, 9

26. E SSENTIAL Q UESTIONS How do I use properties of trapezoids? How do I use properties of kites?

27. V OCABULARY Trapezoid – a quadrilateral with exactly one pair of parallel sides. A trapezoid has two pairs of base angles. In this example the base angles are  A &  B and  C &  D leg base

28. 5.3 I SOSCELES T RAPEZOID C ONJECTURE If a trapezoid is isosceles, then each pair of base angles is congruent.  A   B,  C   D

29. 5.3 I SOSCELES T RAPEZOID C ONJECTURE C ONVERSE If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. ABCD is an isosceles trapezoid

30. 8.16 D IAGONALS OF A T RAPEZOID C ONJECTURE A trapezoid is isosceles if and only if its diagonals are congruent.

31. E XAMPLE 1 PQRS is an isosceles trapezoid. Find m  P, m  Q and m  R. m  R = 50  since base angles are congruent m  P = 130  and m  Q = 130  (consecutive angles of parallel lines cut by a transversal are  )

32. E X. 2: U SING PROPERTIES OF TRAPEZOIDS Show that ABCD is a trapezoid. Compare the slopes of opposite sides. The slope of AB = 5 – 0 = 5 = - 1 0 – 5 -5 The slope of CD = 4 – 7 = -3 = - 1 7 – 4 3 The slopes of AB and CD are equal, so AB ║ CD. The slope of BC = 7 – 5 = 2 = 1 4 – 0 4 2 The slope of AD = 4 – 0 = 4 = 2 7 – 5 2 The slopes of BC and AD are not equal, so BC is not parallel to AD. So, because AB ║ CD and BC is not parallel to AD, ABCD is a trapezoid.

33. D EFINITION Kite – a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

34. 8.19 T HEOREM : O PPOSITE A NGLES OF A K ITE If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent  A   C,  B   D

35. 8.18 T HEOREM : P ERPENDICULAR D IAGONALS OF A K ITE If a quadrilateral is a kite, then its diagonals are perpendicular.

36. E XAMPLE 2 Find the side lengths of the kite.

37. E XAMPLE 2 C ONTINUED We can use the Pythagorean Theorem to find the side lengths. 12 2 + 20 2 = (WX) 2 144 + 400 = (WX) 2 544 = (WX) 2 12 2 + 12 2 = (XY) 2 144 + 144 = (XY) 2 288 = (XY) 2

38. E XAMPLE 3 Find m  G and m  J. Since GHJK is a kite  G   J So 2(m  G) + 132  + 60  = 360  2(m  G) =168  m  G = 84  and m  J = 84 

39. T RY T HIS ! RSTU is a kite. Find m  R, m  S and m  T. x +30 + 125 + 125 + x = 360 2x + 280 = 360 2x = 80 x = 40 So m  R = 70 , m  T = 40  and m  S = 125 

40. E X. 4: U SING THE DIAGONALS OF A KITE WXYZ is a kite so the diagonals are perpendicular. You can use the Pythagorean Theorem to find the side lengths. WX = √ 20 2 + 12 2 ≈ 23.32 XY = √ 12 2 + 12 2 ≈ 16.97 Because WXYZ is a kite, WZ = WX ≈ 23.32, and ZY = XY ≈ 16.97

41. E X. 5: A NGLES OF A KITE Find m  G and m  J in the diagram at the right. SOLUTION: GHJK is a kite, so  G ≅  J and m  G = m  J. 2(m  G) + 132 ° + 60° = 360°Sum of measures of int.  s of a quad. is 360° 2(m  G) = 168°Simplify m  G = 84° Divide each side by 2. So, m  J = m  G = 84° 132 ° 60 °

42. 5.4 P ROPERTIES OF M IDSEGMENTS D EFINE AND DISCOVER PROPERTIES OF MIDSEGMENTS IN TRIANGLES AND TRAPEZOIDS H OMEWORK Go over class notes, solve pg.278 # 5, 6, 7

45. H OMEWORK

46. D EFINITION Midsegment of a trapezoid – the segment that connects the midpoints of the legs.

47. M IDSEGMENT T HEOREM FOR T RAPEZOIDS The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.

48. E X. 3: F INDING M IDSEGMENT LENGTHS OF TRAPEZOIDS LAYER CAKE A baker is making a cake like the one at the right. The top layer has a diameter of 8 inches and the bottom layer has a diameter of 20 inches. How big should the middle layer be?

49. E X. 3: F INDING M IDSEGMENT LENGTHS OF TRAPEZOIDS Use the midsegment theorem for trapezoids. DG = ½(EF + CH)= ½ (8 + 20) = 14” C D E D G F