The document discusses rhombi and squares. It defines a rhombus as a parallelogram with four congruent sides and gives its properties. A square is defined as a parallelogram with four right angles and four congruent sides. The document provides theorems for identifying rhombi and squares. It then gives examples of using the properties and theorems to determine if a shape is a rhombus, rectangle, or square.

1. Section 6-5
Rhombi and Squares

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?

3. Vocabulary
1. Rhombus:
Properties:
2. Square:
Properties:

4. Vocabulary
1. Rhombus: A parallelogram with four congruent sides
Properties:
2. Square:
Properties:

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:

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:

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

8. Theorems
Diagonals of a Rhombus
6.15:
6.16:

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

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

11. Theorems
Conditions for Rhombi and Squares
6.17:
6.18:
6.19:

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:

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:

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

15. Theorems
Conditions for Rhombi and Squares
6.20:

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

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.

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°.

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°

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°

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°

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.

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

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

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

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

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

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

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

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

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

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

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 Thm.
2. LM ∣∣ PN and MN ∣∣ PL

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 Thm.
4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5
2. LM ∣∣ PN and MN ∣∣ PL

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 Thm.
4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5 4. Transitive
2. LM ∣∣ PN and MN ∣∣ PL

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 Thm.
4. ∠1 ≅ ∠ 6 and ∠6 ≅ ∠5 4. Transitive
5. LMNP is a rhombus
2. LM ∣∣ PN and MN ∣∣ PL

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 Thm.
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

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?

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.