Rhombi and Squares

1. Section 6-5
Rhombi and Squares
Tuesday, April 29, 14

2. Essential Questions
• How do you recognize and apply the properties of
rhombi and squares?
• How do you determine whether quadrilaterals are
rectangles, rhombi, or squares?
Tuesday, April 29, 14

3. Vocabulary
1. Rhombus:
Properties:
2. Square:
Properties:
Tuesday, April 29, 14

4. Vocabulary
1. Rhombus: A parallelogram with four congruent sides
Properties:
2. Square:
Properties:
Tuesday, April 29, 14

5. Vocabulary
1. Rhombus: A parallelogram with four congruent sides
Properties: All properties of parallelograms plus
four congruent sides, perpendicular diagonals, and
angles that are bisected by the diagonals
2. Square:
Properties:
Tuesday, April 29, 14

6. Vocabulary
1. Rhombus: A parallelogram with four congruent sides
Properties: All properties of parallelograms plus
four congruent sides, perpendicular diagonals, and
angles that are bisected by the diagonals
2. Square: A parallelogram with four congruent sides
and four right angles
Properties:
Tuesday, April 29, 14

7. Vocabulary
1. Rhombus: A parallelogram with four congruent sides
Properties: All properties of parallelograms plus
four congruent sides, perpendicular diagonals, and
angles that are bisected by the diagonals
2. Square: A parallelogram with four congruent sides
and four right angles
Properties: All properties of parallelograms,
rectangles, and rhombi
Tuesday, April 29, 14

8. Theorems
Diagonals of a Rhombus
6.15:
6.16:
Tuesday, April 29, 14

9. Theorems
Diagonals of a Rhombus
6.15: If a parallelogram is a rhombus, then its
diagonals are perpendicular
6.16:
Tuesday, April 29, 14

10. Theorems
Diagonals of a Rhombus
6.15: If a parallelogram is a rhombus, then its
diagonals are perpendicular
6.16: If a parallelogram is a rhombus, then each
diagonal bisects a pair of opposite angles
Tuesday, April 29, 14

11. Theorems
Conditions for Rhombi and Squares
6.17:
6.18:
6.19:
Tuesday, April 29, 14

12. Theorems
Conditions for Rhombi and Squares
6.17: If the diagonals of a parallelogram are
perpendicular, then the parallelogram is a rhombus
6.18:
6.19:
Tuesday, April 29, 14

13. Theorems
Conditions for Rhombi and Squares
6.17: If the diagonals of a parallelogram are
perpendicular, then the parallelogram is a rhombus
6.18: If one diagonal of a parallelogram bisects a
pair of opposite angles, then the parallelogram is a
rhombus
6.19:
Tuesday, April 29, 14

14. Theorems
Conditions for Rhombi and Squares
6.17: If the diagonals of a parallelogram are
perpendicular, then the parallelogram is a rhombus
6.18: If one diagonal of a parallelogram bisects a
pair of opposite angles, then the parallelogram is a
rhombus
6.19: If one pair of consecutive sides of a
parallelogram are congruent, then the parallelogram
is a rhombus
Tuesday, April 29, 14

15. Theorems
Conditions for Rhombi and Squares
6.20:
Tuesday, April 29, 14

16. Theorems
Conditions for Rhombi and Squares
6.20: If a quadrilateral is both a rectangle and a
rhombus, then it is a square
Tuesday, April 29, 14

17. Example 1
The diagonals of a rhombus WXYZ intersect at V. Use
the given information to find each measure or value.
a. If m∠WZX = 39.5°, find m∠ZYX.
Tuesday, April 29, 14

18. Example 1
The diagonals of a rhombus WXYZ intersect at V. Use
the given information to find each measure or value.
a. If m∠WZX = 39.5°, find m∠ZYX.
Since m∠WZX = 39.5°, we know that
m∠WZY = 79°.
Tuesday, April 29, 14

19. Example 1
The diagonals of a rhombus WXYZ intersect at V. Use
the given information to find each measure or value.
a. If m∠WZX = 39.5°, find m∠ZYX.
Since m∠WZX = 39.5°, we know that
m∠WZY = 79°.
180° − 79°
Tuesday, April 29, 14

20. Example 1
The diagonals of a rhombus WXYZ intersect at V. Use
the given information to find each measure or value.
a. If m∠WZX = 39.5°, find m∠ZYX.
Since m∠WZX = 39.5°, we know that
m∠WZY = 79°.
180° − 79° = 101°
Tuesday, April 29, 14

21. Example 1
The diagonals of a rhombus WXYZ intersect at V. Use
the given information to find each measure or value.
a. If m∠WZX = 39.5°, find m∠ZYX.
Since m∠WZX = 39.5°, we know that
m∠WZY = 79°.
180° − 79° = 101°
m∠ZYX = 101°
Tuesday, April 29, 14

22. Example 1
The diagonals of a rhombus WXYZ intersect at V. Use
the given information to find each measure or value.
b. If WX = 8x − 5 and WZ = 6x + 3,
find x.
Tuesday, April 29, 14

23. Example 1
The diagonals of a rhombus WXYZ intersect at V. Use
the given information to find each measure or value.
b. If WX = 8x − 5 and WZ = 6x + 3,
find x.
WX = WZ
Tuesday, April 29, 14

24. Example 1
The diagonals of a rhombus WXYZ intersect at V. Use
the given information to find each measure or value.
b. If WX = 8x − 5 and WZ = 6x + 3,
find x.
WX = WZ
8x − 5 = 6x + 3
Tuesday, April 29, 14

25. Example 1
The diagonals of a rhombus WXYZ intersect at V. Use
the given information to find each measure or value.
b. If WX = 8x − 5 and WZ = 6x + 3,
find x.
WX = WZ
8x − 5 = 6x + 3
2x = 8
Tuesday, April 29, 14

26. Example 1
The diagonals of a rhombus WXYZ intersect at V. Use
the given information to find each measure or value.
b. If WX = 8x − 5 and WZ = 6x + 3,
find x.
WX = WZ
8x − 5 = 6x + 3
2x = 8
x = 4
Tuesday, April 29, 14

27. Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and
∠2 ≅ ∠6
Prove: LMNP is a rhombus
Tuesday, April 29, 14

28. Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and
∠2 ≅ ∠6
Prove: LMNP is a rhombus
1. LMNP is a parallelogram,
∠1 ≅ ∠2, and ∠2 ≅ ∠6
Tuesday, April 29, 14

29. Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and
∠2 ≅ ∠6
Prove: LMNP is a rhombus
1. LMNP is a parallelogram,
∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
Tuesday, April 29, 14

30. Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and
∠2 ≅ ∠6
Prove: LMNP is a rhombus
1. LMNP is a parallelogram,
∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
2. LM ∣∣ PN and MN ∣∣ PL
Tuesday, April 29, 14

31. Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and
∠2 ≅ ∠6
Prove: LMNP is a rhombus
1. LMNP is a parallelogram,
∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
2. Def. of parallelogram2. LM ∣∣ PN and MN ∣∣ PL
Tuesday, April 29, 14

32. Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and
∠2 ≅ ∠6
Prove: LMNP is a rhombus
1. LMNP is a parallelogram,
∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
2. Def. of parallelogram
3. ∠5 ≅ ∠1
2. LM ∣∣ PN and MN ∣∣ PL
Tuesday, April 29, 14

33. Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and
∠2 ≅ ∠6
Prove: LMNP is a rhombus
1. LMNP is a parallelogram,
∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
2. Def. of parallelogram
3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅
2. LM ∣∣ PN and MN ∣∣ PL
Tuesday, April 29, 14

34. Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and
∠2 ≅ ∠6
Prove: LMNP is a rhombus
1. LMNP is a parallelogram,
∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
2. Def. of parallelogram
3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅
4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5
2. LM ∣∣ PN and MN ∣∣ PL
Tuesday, April 29, 14

35. Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and
∠2 ≅ ∠6
Prove: LMNP is a rhombus
1. LMNP is a parallelogram,
∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
2. Def. of parallelogram
3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅
4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5 4. Transitive
2. LM ∣∣ PN and MN ∣∣ PL
Tuesday, April 29, 14

36. Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and
∠2 ≅ ∠6
Prove: LMNP is a rhombus
1. LMNP is a parallelogram,
∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
2. Def. of parallelogram
3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅
4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5 4. Transitive
5. LMNP is a rhombus
2. LM ∣∣ PN and MN ∣∣ PL
Tuesday, April 29, 14

37. Example 2
Given: LMNP is a parallelogram, ∠1 ≅ ∠2, and
∠2 ≅ ∠6
Prove: LMNP is a rhombus
1. LMNP is a parallelogram,
∠1 ≅ ∠2, and ∠2 ≅ ∠6
1. Given
2. Def. of parallelogram
3. ∠5 ≅ ∠1 3. Alt. int. ∠’s of ∣∣ lines ≅
4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5 4. Transitive
5. LMNP is a rhombus 5. One diagonal bisects
opposite angles
2. LM ∣∣ PN and MN ∣∣ PL
Tuesday, April 29, 14

38. Example 3
Matt Mitarnowski is measuring the boundary of a
new garden. He wants the garden to be square. He
has set each of the corner stakes 6 feet apart.
What does Matt need to know to make sure that
the garden is a square?
Tuesday, April 29, 14

39. Example 3
Matt Mitarnowski is measuring the boundary of a
new garden. He wants the garden to be square. He
has set each of the corner stakes 6 feet apart.
What does Matt need to know to make sure that
the garden is a square?
Matt knows he at least has a rhombus as all four
sides are congruent. To make it a square, it must
also be a rectangle, so the diagonals must also be
congruent, as anything that is both a rhombus and
a rectangle is also a square.
Tuesday, April 29, 14

40. Example 4
Determine whether parallelogram ABCD is a
rhombus, rectangle, or square for A(−2, −1), B(−1, 3),
C(3, 2), and D(2, −2). List all that apply and
explain how you know.
Tuesday, April 29, 14

41. Example 4
Determine whether parallelogram ABCD is a
rhombus, rectangle, or square for A(−2, −1), B(−1, 3),
C(3, 2), and D(2, −2). List all that apply and
explain how you know.
AC = (−2 − 3)2
+ (−1− 2)2
Tuesday, April 29, 14

42. Example 4
Determine whether parallelogram ABCD is a
rhombus, rectangle, or square for A(−2, −1), B(−1, 3),
C(3, 2), and D(2, −2). List all that apply and
explain how you know.
AC = (−2 − 3)2
+ (−1− 2)2
= (−5)2
+ (−3)2
Tuesday, April 29, 14

43. Example 4
Determine whether parallelogram ABCD is a
rhombus, rectangle, or square for A(−2, −1), B(−1, 3),
C(3, 2), and D(2, −2). List all that apply and
explain how you know.
AC = (−2 − 3)2
+ (−1− 2)2
= (−5)2
+ (−3)2
= 25 + 9
Tuesday, April 29, 14

44. Example 4
Determine whether parallelogram ABCD is a
rhombus, rectangle, or square for A(−2, −1), B(−1, 3),
C(3, 2), and D(2, −2). List all that apply and
explain how you know.
AC = (−2 − 3)2
+ (−1− 2)2
= (−5)2
+ (−3)2
= 25 + 9 = 34
Tuesday, April 29, 14

45. Example 4
Determine whether parallelogram ABCD is a
rhombus, rectangle, or square for A(−2, −1), B(−1, 3),
C(3, 2), and D(2, −2). List all that apply and
explain how you know.
AC = (−2 − 3)2
+ (−1− 2)2
= (−5)2
+ (−3)2
= 25 + 9 = 34
BD = (−1− 2)2
+ (3 + 2)2
Tuesday, April 29, 14

46. Example 4
Determine whether parallelogram ABCD is a
rhombus, rectangle, or square for A(−2, −1), B(−1, 3),
C(3, 2), and D(2, −2). List all that apply and
explain how you know.
AC = (−2 − 3)2
+ (−1− 2)2
= (−5)2
+ (−3)2
= 25 + 9 = 34
BD = (−1− 2)2
+ (3 + 2)2
= (−3)2
+ (5)2
Tuesday, April 29, 14

47. Example 4
Determine whether parallelogram ABCD is a
rhombus, rectangle, or square for A(−2, −1), B(−1, 3),
C(3, 2), and D(2, −2). List all that apply and
explain how you know.
AC = (−2 − 3)2
+ (−1− 2)2
= (−5)2
+ (−3)2
= 25 + 9 = 34
BD = (−1− 2)2
+ (3 + 2)2
= (−3)2
+ (5)2
= 9 + 25
Tuesday, April 29, 14

48. Example 4
Determine whether parallelogram ABCD is a
rhombus, rectangle, or square for A(−2, −1), B(−1, 3),
C(3, 2), and D(2, −2). List all that apply and
explain how you know.
AC = (−2 − 3)2
+ (−1− 2)2
= (−5)2
+ (−3)2
= 25 + 9 = 34
BD = (−1− 2)2
+ (3 + 2)2
= (−3)2
+ (5)2
= 9 + 25 = 34
Tuesday, April 29, 14

49. Example 4
Determine whether parallelogram ABCD is a
rhombus, rectangle, or square for A(−2, −1), B(−1, 3),
C(3, 2), and D(2, −2). List all that apply and
explain how you know.
AC = (−2 − 3)2
+ (−1− 2)2
= (−5)2
+ (−3)2
= 25 + 9 = 34
BD = (−1− 2)2
+ (3 + 2)2
= (−3)2
+ (5)2
= 9 + 25 = 34
The diagonals are congruent, so it is a rectangle
Tuesday, April 29, 14

50. Example 4
Determine whether parallelogram ABCD is a
rhombus, rectangle, or square for A(−2, −1), B(−1, 3),
C(3, 2), and D(2, −2). List all that apply and
explain how you know.
AC = (−2 − 3)2
+ (−1− 2)2
= (−5)2
+ (−3)2
= 25 + 9 = 34
BD = (−1− 2)2
+ (3 + 2)2
= (−3)2
+ (5)2
= 9 + 25 = 34
The diagonals are congruent, so it is a rectangle
m( AC) =
2 +1
3 + 2
Tuesday, April 29, 14

51. Example 4
Determine whether parallelogram ABCD is a
rhombus, rectangle, or square for A(−2, −1), B(−1, 3),
C(3, 2), and D(2, −2). List all that apply and
explain how you know.
AC = (−2 − 3)2
+ (−1− 2)2
= (−5)2
+ (−3)2
= 25 + 9 = 34
BD = (−1− 2)2
+ (3 + 2)2
= (−3)2
+ (5)2
= 9 + 25 = 34
The diagonals are congruent, so it is a rectangle
m( AC) =
2 +1
3 + 2
=
3
5
Tuesday, April 29, 14

52. Example 4
Determine whether parallelogram ABCD is a
rhombus, rectangle, or square for A(−2, −1), B(−1, 3),
C(3, 2), and D(2, −2). List all that apply and
explain how you know.
AC = (−2 − 3)2
+ (−1− 2)2
= (−5)2
+ (−3)2
= 25 + 9 = 34
BD = (−1− 2)2
+ (3 + 2)2
= (−3)2
+ (5)2
= 9 + 25 = 34
The diagonals are congruent, so it is a rectangle
m( AC) =
2 +1
3 + 2
=
3
5
m(BD) =
−2 − 3
2 +1
Tuesday, April 29, 14

53. Example 4
Determine whether parallelogram ABCD is a
rhombus, rectangle, or square for A(−2, −1), B(−1, 3),
C(3, 2), and D(2, −2). List all that apply and
explain how you know.
AC = (−2 − 3)2
+ (−1− 2)2
= (−5)2
+ (−3)2
= 25 + 9 = 34
BD = (−1− 2)2
+ (3 + 2)2
= (−3)2
+ (5)2
= 9 + 25 = 34
The diagonals are congruent, so it is a rectangle
m( AC) =
2 +1
3 + 2
=
3
5
m(BD) =
−2 − 3
2 +1
=
−5
3
Tuesday, April 29, 14

54. Example 4
Determine whether parallelogram ABCD is a
rhombus, rectangle, or square for A(−2, −1), B(−1, 3),
C(3, 2), and D(2, −2). List all that apply and
explain how you know.
AC = (−2 − 3)2
+ (−1− 2)2
= (−5)2
+ (−3)2
= 25 + 9 = 34
BD = (−1− 2)2
+ (3 + 2)2
= (−3)2
+ (5)2
= 9 + 25 = 34
The diagonals are congruent, so it is a rectangle
m( AC) =
2 +1
3 + 2
=
3
5
m(BD) =
−2 − 3
2 +1
=
−5
3
AC ⊥ BD
Tuesday, April 29, 14

55. Example 4
Determine whether parallelogram ABCD is a
rhombus, rectangle, or square for A(−2, −1), B(−1, 3),
C(3, 2), and D(2, −2). List all that apply and
explain how you know.
AC = (−2 − 3)2
+ (−1− 2)2
= (−5)2
+ (−3)2
= 25 + 9 = 34
BD = (−1− 2)2
+ (3 + 2)2
= (−3)2
+ (5)2
= 9 + 25 = 34
The diagonals are congruent, so it is a rectangle
m( AC) =
2 +1
3 + 2
=
3
5
m(BD) =
−2 − 3
2 +1
=
−5
3
AC ⊥ BD
The diagonals are ⊥, so it is a rhombus, thus also a square.
Tuesday, April 29, 14

56. Problem Set
Tuesday, April 29, 14

57. Problem Set
p. 431 #1-33 odd, 46, 55, 57, 59
"Courage is saying, 'Maybe what I'm doing isn't
working; maybe I should try something else.'"
- Anna Lappe
Tuesday, April 29, 14