LESSON-4-Trapezoids-and-Kites Grade -9 3

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1. LESSON 4 TRAPEZOIDS AND KITES

2. Objectives: The learners will be able to: Define and illustrate trapezoid, isosceles trapezoid and kites prove theorems on trapezoids and kites. solve problems involving trapezoids and kites.

3. T R A P E Z O I D

4. If only one of two pairs of opposite side is parallel, the quadrilateral is called a trapezoid . R E M E M B E R :

5. Definition of Trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel sides.   T Q S R T Q S R base base leg leg Base angles The parallel sides are called the bases of the trapezoid . The nonparallel sides are called legs . The base and the legs are called base angles.

6. Example 1: Quadrilateral ABCD is a trapezoid. D A C B a. Name the legs of the given quadrilateral. Answer: AD and BC

7. D A C B b. Name the bases of the given quadrilateral. Answer: and  

8. D A C B c. Name the angles formed by its legs and bases. Answer: and  

9. The following illustrations show that the properties that distinguish parallelogram do not apply to all trapezoids. A B C D   E F G H   *The diagonals of a trapezoid may or may not be congruent.

10. E F G H   I n trapezoid EFGH, the diagonals are not the only pair of segments that seems congruent.

11. E F G H   The legs and the base angles of this trapezoid seem congruent too. This kind of trapezoid is referred to as isosceles trapezoid . Note:

12. Definition of Isosceles Trapezoid An isosceles trapezoid is a trapezoid with congruent legs.   J K L M

13. AC≅BD PROPERTIES OF AN ISOSCELES TRAPEZOID The legs are congruent. Each pair of base angles are congruent. Diagonals are congruent

14. AC≅BD PROPERTIES OF AN ISOSCELES TRAPEZOID The median of a trapezoid is parallel to the bases. The length of the median is one-half of the sum of the lengths of the two bases. The sum of all the angles of a trapezoid is 360°.

15. CONDITIONS FOR A TRAPEZOID TO BE ISOSCELES If a trapezoid has a pair of congruent base angles, then the trapezoid is isosceles. If the diagonals of a trapezoid are congruent, then the trapezoid is isosceles.

16. Example 2: GEOM is an isosceles trapezoid. Find the , , and   G E O M  

17. SOLUTION: GEOM is a trapezoid, so m∠G = m∠E =48°. ∠G and ∠M are consecutive angles, so m∠G+m∠M =180. G E O M  

18.         G E O M  

19. B A C D Example 3: In an isosceles trapezoid ABCD, m∠A =3x+40 and m∠D =x+60 . Find ∠B .

20. B A C D SOLUTION: Find the value of variable x. By definition.  

21. B A C D SOLUTION: Substitute the obtained value of x in ∠A.  

22. B A C D SOLUTION: With as transversal to parallel and , and are supplementary .    

23. Definition of Median of a Trapezoid The median of a trapezoid is the segment joining the midpoints of the legs. M P N O Q R In trapezoid MNOP, is the median . Note that the median of a trapezoid is parallel to its base.  

24. Theorem 99 The median of a trapezoid is parallel to its base. Theorem 100 The median of a trapezoid is half the sum of the lengths of the bases.

25. Example 4: In the figure below, PQRS is an isosceles trapezoid with PQ∥SR , and XY are midpoints of PS and QR , respectively P S Q R X Y

26. P S Q R X Y Name the legs of trapezoid PQRS Answer: and  

27. P S Q R X Y b. Name the bases of trapezoid PQRS Answer: and  

28. P S Q R X Y c. Name the median of trapezoid PQRS Answer:  

29. P S Q R X Y d . Name the congruent angles of trapezoid PQRS Answer: and  

30. Example 5: The lengths of the bases of a trapezoid are 17 cm and 11 cm. how long is its median?

31. SOLUTIO N :   Thus, the median is 28 cm ÷ 2 = 14 cm.

32.   Example 6: ABCD is a trapezoid, with as median. If EF = 2.3 and AB = 4, find DC.   A B C D E F

33.   SOLUTION:   A B C D E F

34. T W U V A B Example 7: TUVW is a trapezoid. A and B are midpoints of TW and UV , respectively. what is the length of AB if TU = 6 cm and WV = 11 cm?

35. T W U V A B SOLUTION:  

36. Example 8: ABCD is a trapezoid with median EF . D A C B E F If DC = 2x + 10, AB = 3x + 20, and EF = 4x, what is EF?

37. SOLUTION: a. By Theorem 100   D A C B E F  

38. Example 8: ABCD is a trapezoid with median EF . D A C B E F b. If AB = 4x – 6, EF = 3x, and DC = 7(x-2), what is DC?

39. D A C B E F b. By Theorem 100   SOLUTION:  

40. K I T E

41. A kite is another type of a quadrilateral. It has two distinct pairs of adjacent congruent sides. Quadrilateral ABCD on the right side is a kite.

42. A B C D In kite ABCD, the two congruent sides are: and two segments that are diagonals are: and  

43. Theorem 101 If a quadrilateral is a kite, then its diagonals are perpendicular. E F G H Quadrilateral EFGH is a kite. Its diagonal must be perpendicular.

44. The converse of Theorem 101 is stated in the next theorem. One interesting property of kites has something to do with its area.

45. J K L M Quadrilateral JKLM is a kite. The lengths of the sides are congruent, and their diagonals are perpendicular to one another.

46. Theorem 102 If exactly one diagonal of a quadrilateral is the perpendicular bisector of the other diagonal, then the quadrilateral is a kite. Theorem 103 The area of a kite is half the product of the lengths of its diagonals.

47. Example 9: Given: kite MNOP If NP = 2.5 cm and MO = 4.5 cm, find the area of kite MNOP. M P O N

48. SOLUTION: By Theorem 103 Area of kite   M P O N

49. Example 10: Given: kite QRST with area 27 cm 2 . Find TR if QS = 4.5 cm. Q T S R

50. SOLUTION: By Theorem 103 Area of kite   Q T S R

51. Example 11: Given: kite ABLE Find: What is the value of x?  

52. SOLUTION (a):  

53. SOLUTION (b):  

54. SOLUTION (c):