1. 18/02/2014 CH.6.5 Conditions for Special Parallelogram

2. Classwork Homework Homework Booklet Chapter: 6.5
(Pages 434 to 436 ) Exercises 1, 4, 5, 6, 9 to 16, 17, 18, 20, 22, 24 to 26, 27, 28, 33(a, b, c), 34, 39, 40, 41, 43.
Homework Booklet
Chapter: 6.5

3. Warm Up
Solve for x.
1. 16x – 3 = 12x + 13
2. 2x – 4 = 90
ABCD is a parallelogram. Find each measure.
3. CD 4. mC 4 47 14
104°

4. Warm Up 5 –1 1. Find AB for A (–3, 5) and B (1, 2).
2. Find the slope of JK for J(–4, 4) and K(3, –3).
ABCD is a parallelogram. Justify each statement.
3. ABC  CDA
4. AEB  CED 5 –1
 opp. s 
Vert. s Thm.

5. Objective
Prove that a given quadrilateral is a rectangle, rhombus, or square.

6. When you are given a parallelogram with certain
properties, you can use the theorems below to determine whether the parallelogram is a rectangle.

7. Example 1: Carpentry Application
A manufacture builds a mold for a desktop so that , , and mABC = 90°. Why must ABCD be a rectangle?
Both pairs of opposites sides of ABCD are congruent, so ABCD is a . Since mABC = 90°, one angle ABCD is a right angle. ABCD is a rectangle by Theorem

8. Check It Out! Example 1
A carpenter’s square can be used to test that an angle is a right angle. How could the contractor use a carpenter’s square to check that the frame is a rectangle?
Both pairs of opp. sides of WXYZ are , so WXYZ is a parallelogram. The contractor can use the carpenter’s square to see if one  of WXYZ is a right . If one angle is a right , then by Theorem the frame is a rectangle.

9. Below are some conditions you can use to determine whether a parallelogram is a rhombus.

10. In order to apply Theorems 6-5-1 through 6-5-5, the quadrilateral must be a parallelogram.
Caution
To prove that a given quadrilateral is a square, it is sufficient to show that the figure is both a rectangle and a rhombus. You will explain why this is true in Exercise 43.

11. You can also prove that a given quadrilateral is a
rectangle, rhombus, or square by using the definitions of the special quadrilaterals.
Remember!

12. Example 2A: Applying Conditions for Special Parallelograms
Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid.
Given:
Conclusion: EFGH is a rhombus.
The conclusion is not valid. By Theorem 6-5-3, if one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. By Theorem 6-5-4, if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. To apply either theorem, you must first know that ABCD is a parallelogram.

13. Example 2B: Applying Conditions for Special Parallelograms
Determine if the conclusion is valid.
If not, tell what additional information is needed to make it valid.
Given:
Conclusion: EFGH is a square.
Step 1 Determine if EFGH is a parallelogram.
Given
Quad. with diags. bisecting each other 
EFGH is a parallelogram.

14. Example 2B Continued
Step 2 Determine if EFGH is a rectangle.
Given.
EFGH is a rectangle.
with diags.   rect.
Step 3 Determine if EFGH is a rhombus.
with one pair of cons. sides   rhombus
EFGH is a rhombus.

15. Example 2B Continued
Step 4 Determine is EFGH is a square.
Since EFGH is a rectangle and a rhombus, it has four right angles and four congruent sides. So EFGH is a square by definition.
The conclusion is valid.

16. Check It Out! Example 2
Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid.
Given: ABC is a right angle.
Conclusion: ABCD is a rectangle.
The conclusion is not valid. By Theorem 6-5-1, if one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. To apply this theorem, you need to know that ABCD is a parallelogram .

17. Example 3A: Identifying Special Parallelograms in the Coordinate Plane
Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply.
P(–1, 4), Q(2, 6), R(4, 3), S(1, 1)

18. Example 3A Continued
Step 1 Graph PQRS.

19. Example 3A Continued
Step 2 Find PR and QS to determine if PQRS is a rectangle.
Since , the diagonals are congruent. PQRS is a rectangle.

20. Example 3A Continued
Step 3 Determine if PQRS is a rhombus.
Since , PQRS is a rhombus.
Step 4 Determine if PQRS is a square.
Since PQRS is a rectangle and a rhombus, it has four right angles and four congruent sides. So PQRS is a square by definition.

21. Example 3B: Identifying Special Parallelograms in the Coordinate Plane
Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply.
W(0, 1), X(4, 2), Y(3, –2), Z(–1, –3)
Step 1 Graph WXYZ.

22. Example 3B Continued
Step 2 Find WY and XZ to determine if WXYZ is a rectangle.
Since , WXYZ is not a rectangle.
Thus WXYZ is not a square.

23. Example 3B Continued
Step 3 Determine if WXYZ is a rhombus.
Since (–1)(1) = –1, , WXYZ is a rhombus.

24. Check It Out! Example 3A
Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply.
K(–5, –1), L(–2, 4), M(3, 1), N(0, –4)

25. Check It Out! Example 3A Continued
Step 1 Graph KLMN.

26. Check It Out! Example 3A Continued
Step 2 Find KM and LN to determine if KLMN is a rectangle.
Since , KMLN is a rectangle.

27. Check It Out! Example 3A Continued
Step 3 Determine if KLMN is a rhombus.
Since the product of the slopes is –1, the two lines are perpendicular. KLMN is a rhombus.

28. Check It Out! Example 3A Continued
Step 4 Determine if KLMN is a square.
Since KLMN is a rectangle and a rhombus, it has four right angles and four congruent sides. So KLMN is a square by definition.

29. Check It Out! Example 3B
Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply.
P(–4, 6) , Q(2, 5) , R(3, –1) , S(–3, 0)

30. Check It Out! Example 3B Continued
Step 1 Graph PQRS.

31. Check It Out! Example 3B Continued
Step 2 Find PR and QS to determine if PQRS is a rectangle.
Since , PQRS is not a rectangle. Thus PQRS is not a square.

32. Check It Out! Example 3B Continued
Step 3 Determine if PQRS is a rhombus.
Since (–1)(1) = –1, are perpendicular and congruent. PQRS is a rhombus.

33. Lesson Quiz: Part I
1. Given that AB = BC = CD = DA, what additional information is needed to conclude that ABCD is a square?

34. Lesson Quiz: Part II
2. Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid.
Given: PQRS and PQNM are parallelograms.
Conclusion: MNRS is a rhombus.
valid

35. Lesson Quiz: Part III
3. Use the diagonals to determine whether a parallelogram with vertices A(2, 7), B(7, 9), C(5, 4), and D(0, 2) is a rectangle, rhombus, or square. Give all the names that apply.
AC ≠ BD, so ABCD is not a rect. or a square. The slope of AC = –1, and the slope of BD
= 1, so AC  BD. ABCD is a rhombus.

36. Objectives
Prove and apply properties of rectangles, rhombuses, and squares.
Use properties of rectangles, rhombuses, and squares to solve problems.

37. Vocabulary
rectangle
rhombus
square

38. A second type of special quadrilateral is a rectangle
A second type of special quadrilateral is a rectangle. A rectangle is a quadrilateral with four right angles.

39. Since a rectangle is a parallelogram by Theorem 6-4-1, a rectangle “inherits” all the properties of parallelograms that you learned in Lesson 6-2.

40. Example 1: Craft Application
A woodworker constructs a rectangular picture frame so that JK = 50 cm and JL = 86 cm. Find HM.
Rect.  diags. 
KM = JL = 86
Def. of  segs.
 diags. bisect each other
Substitute and simplify.

41. Check It Out! Example 1a
Carpentry The rectangular gate has diagonal braces.
Find HJ.
Rect.  diags. 
HJ = GK = 48
Def. of  segs.

42. Check It Out! Example 1b
Carpentry The rectangular gate has diagonal braces.
Find HK.
Rect.  diags. 
Rect.  diagonals bisect each other
JL = LG
Def. of  segs.
JG = 2JL = 2(30.8) = 61.6
Substitute and simplify.

43. A rhombus is another special quadrilateral
A rhombus is another special quadrilateral. A rhombus is a quadrilateral with four congruent sides.

45. Like a rectangle, a rhombus is a parallelogram
Like a rectangle, a rhombus is a parallelogram. So you can apply the properties of parallelograms to rhombuses.

46. Example 2A: Using Properties of Rhombuses to Find Measures
TVWX is a rhombus. Find TV.
WV = XT
Def. of rhombus
13b – 9 = 3b + 4
Substitute given values.
10b = 13
Subtract 3b from both sides and add 9 to both sides.
b = 1.3
Divide both sides by 10.

47. Example 2A Continued
TV = XT
Def. of rhombus
Substitute 3b + 4 for XT.
TV = 3b + 4
TV = 3(1.3) + 4 = 7.9
Substitute 1.3 for b and simplify.

48. Example 2B: Using Properties of Rhombuses to Find Measures
TVWX is a rhombus. Find mVTZ.
mVZT = 90°
Rhombus  diag. 
14a + 20 = 90°
Substitute 14a + 20 for mVTZ.
Subtract 20 from both sides and divide both sides by 14.
a = 5

49. Example 2B Continued
Rhombus  each diag. bisects opp. s
mVTZ = mZTX
mVTZ = (5a – 5)°
Substitute 5a – 5 for mVTZ.
mVTZ = [5(5) – 5)]°
= 20°
Substitute 5 for a and simplify.

50. Check It Out! Example 2a
CDFG is a rhombus. Find CD.
CG = GF
Def. of rhombus
5a = 3a + 17
Substitute
a = 8.5
Simplify
GF = 3a + 17 = 42.5
Substitute
CD = GF
Def. of rhombus
CD = 42.5
Substitute

51. Check It Out! Example 2b
CDFG is a rhombus.
Find the measure.
mGCH if mGCD = (b + 3)°
and mCDF = (6b – 40)°
mGCD + mCDF = 180°
Def. of rhombus
b b – 40 = 180°
Substitute.
7b = 217°
Simplify.
b = 31°
Divide both sides by 7.

52. Check It Out! Example 2b Continued
mGCH + mHCD = mGCD
Rhombus  each diag. bisects opp. s
2mGCH = mGCD
2mGCH = (b + 3)
Substitute.
2mGCH = (31 + 3)
Substitute.
mGCH = 17°
Simplify and divide both sides by 2.

53. A square is a quadrilateral with four right angles and four congruent sides. In the exercises, you will show that a square is a parallelogram, a rectangle, and a rhombus. So a square has the properties of all three.

54. Rectangles, rhombuses, and squares are sometimes referred to as special parallelograms.
Helpful Hint

55. Example 3: Verifying Properties of Squares
Show that the diagonals of square EFGH are congruent perpendicular bisectors of each other.

56. Example 3 Continued
Step 1 Show that EG and FH are congruent.
Since EG = FH,

57. Example 3 Continued
Step 2 Show that EG and FH are perpendicular.
Since ,

58. Example 3 Continued
Step 3 Show that EG and FH are bisect each other.
Since EG and FH have the same midpoint, they bisect each other.
The diagonals are congruent perpendicular bisectors of each other.

59. SV ^ TW SV = TW = 122 so, SV @ TW . slope of TW = –11 1 slope of SV =
Check It Out! Example 3
The vertices of square STVW are S(–5, –4), T(0, 2), V(6, –3) , and W(1, –9) . Show that the diagonals of square STVW are congruent perpendicular bisectors of each other.
SV = TW = so, TW . 1 11
slope of SV =
slope of TW = –11
SV ^ TW

60. Check It Out! Example 3 Continued
Step 1 Show that SV and TW are congruent.
Since SV = TW,

61. Check It Out! Example 3 Continued
Step 2 Show that SV and TW are perpendicular.
Since

62. Check It Out! Example 3 Continued
Step 3 Show that SV and TW bisect each other.
Since SV and TW have the same midpoint, they bisect each other.
The diagonals are congruent perpendicular bisectors of each other.

63. Example 4: Using Properties of Special Parallelograms in Proofs
Given: ABCD is a rhombus. E is the midpoint of , and F is the midpoint of .
Prove: AEFD is a parallelogram.

64. Example 4 Continued ||

65. Check It Out! Example 4
Given: PQTS is a rhombus with diagonal
Prove:

66. Check It Out! Example 4 Continued
Statements
Reasons
1. PQTS is a rhombus.
1. Given.
2. Rhombus → each
diag. bisects opp. s 2.
3. QPR  SPR
3. Def. of  bisector. 4.
4. Def. of rhombus. 5.
5. Reflex. Prop. of  6.
6. SAS 7.
7. CPCTC

67. Lesson Quiz: Part I
A slab of concrete is poured with diagonal spacers. In rectangle CNRT, CN = 35 ft, and NT = 58 ft. Find each length.
1. TR CE
35 ft
29 ft

68. Lesson Quiz: Part II
PQRS is a rhombus. Find each measure.
3. QP mQRP 42
51°

69. Lesson Quiz: Part III
5. The vertices of square ABCD are A(1, 3), B(3, 2), C(4, 4), and D(2, 5). Show that its diagonals are congruent perpendicular bisectors of each other.

70. Lesson Quiz: Part IV
6. Given: ABCD is a rhombus.
Prove: 