1. Unit 6: Lesson #4 Warm-Up:
Pick up the Lesson #4 Example sheet and complete the Warm-Up on the back.

2. Properties of Rhombi, Rectangles, and Squares
Unit 6: Lesson #4
Properties of Rhombi, Rectangles, and Squares
(Textbook Section 6-4)

3. Properties of Rhombi and Squares

4. Statements Reasons Warm-Up: E H G F Given: EFGH is a rhombus
Prove: Diagonal EG is the angle
bisector of E and G
Statements Reasons
EFGH is a rhombus
Given
EF  FG  GH  HE
Definition of a rhombus
EG  EG
Reflexive Property
EFG  EHG
SSS
GEF  GEH and
EGF  EGH
CPCTC
Diagonal EG is the angle bisector of E and G
Definition of an Angle Bisector

5. 1) Could we also prove that diagonal FH is also the angle bisector
of F and H? Explain.
2) Using what we know about isosceles triangles, what can we
conclude about the angle at which the two diagonals meet
each other? Explain. E H G F
Yes, the proof we just completed would have also worked for the diagonal FH. E H G F
The diagonals are each acting as the angle bisectors of the vertex angle of the isosceles triangles. Therefore, they must be perpendicular to each other.

6. Diagonals of a Rhombus Theorem
A parallelogram is a rhombus if and only if the diagonals …
a) bisect the angles of
the rhombus.
b) are perpendicular.

7. NOTE:
Because a rhombus is a parallelogram, the opposite angles are congruent. Therefore, when the diagonals bisect A and C, it creates FOUR congruent angles.
The same thing happens at B and D.

8. Example #1 12 78 12 90 28 90 12 90 28 12 78 124
The following figures are rhombuses. Find the measures of the numbered angles.
78
12
124° 1 2 3
78° 4 1
12
90
28
90
12
90
28 3
12
78 2
124
78 + 90 = 168
180 – 168 = 12
180 – 124 = 56
56 / 2 = 28

9. Properties of Rectangles and Squares

10. Statements Reasons Warm-Up: J M K L Given: JKLM is a rectangle
Prove: Diagonal JL  Diagonal KM
Statements Reasons
JKLM is a rectangle
Given
JM  KL
Opposite sides of a parallelogram are congruent
ML  ML
Reflexive Property
M and L are right angles.
Definition of a rectangle.
M  L
All right angles are congruent
KJM  MLK
SAS
JL  KM
CPCTC

11. Diagonals of a Rectangle Theorem:
A parallelogram is a rectangle if and only if its diagonals are congruent.
NOTE:
Because a rectangle is a parallelogram, the diagonals bisect each other. Therefore, each half of the two diagonals are all congruent to each other.

12. Example #2A A) 1 = ________ BD = ________ 68.5 A D B C 20
137 M
43 20
180 – 137 = 43
AC = BD
180 – 43 = 137
AC = 2 CM = 20
BD = 20
137/2 = 68.5

13. Example #2B
B) Find the length of the diagonals of rectangle QRST if QS = 5y – 9 and RT = y + 5. T Q R S
5y – 9 = y + 5
4y – 9 = 5
4y = 14
y = 3.5

14. Properties of Squares

15. Example #3 4) The figure is a square. EP = 10a – 22 a = ______ FP = 48
y = ______
EP = 10a – 22
FP = 48 7
45° x y E G F H P
90°
(Diagonals of a rhombus are perpendicular)
10a – 22 = 48
10a = 70
a = 7
x = 90 / 2 = 45
(In a rectangle the angles are 90 at the vertices)
(In a rhombus, the diagonals bisect the angles at the vertices)

16. Homework Time!