1. OBJECTIVE: SPECIAL PARALLELOGRAMS
LESSON 6.4
SPECIAL PARALLELOGRAMS
OBJECTIVE:
Use properties of diagonals of rhombuses and rectangles
Determine whether a parallelogram is a rhombus or rectangle

2. Rhombus Theorems Theorem 6.9 Each diagonal of a rhombus
bisects two angles of the rhombus. AC
___ bisects  , so
BAD
1  2 AC
bisects  , so
3  4
BCD

3. Theorem 6.10
The diagonals of a rhombus are
perpendicular.
THEN IF
AC  BD

4. Rectangle Theorem Theorem 6.11 The diagonals of a rectangle are
congruent.
THEN IF
BD  AC

5. Parallelogram Theorems
If one diagonal of a parallelogram bisects two angles of the parallelogram, then
the parallelogram is a rhombus. IF
THEN

6. the parallelogram is a rhombus. IF THEN
Theorem 6.13
If the diagonals of a parallelogram
are perpendicular, then
the parallelogram is a rhombus. IF A B C D
THEN A B C D

7. If the diagonals of a parallelogram are congruent, then
Theorem 6.14
If the diagonals of a parallelogram are congruent, then
the parallelogram is a rectangle. IF
THEN A B C D B A C D
AC  BD

8. diagonals of a rhombus bisect the ’s.
EXAMPLE #1
Find the measures of the numbered angles in the rhombus.
m1 = ____, Why?
78
diagonals of a rhombus bisect the ’s.

9. m2 =
90
Why?
Diagonals of a rhombus are .
m3 =
12
Why?
Triangle sum.
AIA are 
m4 =
78
Why?

10. . The diagonals of a rectangle are 8x + 2 = 5x + 11 Diagonal = 3x = 9
EXAMPLE #2
One diagonal of a rectangle has length 8x The other has length 5x Find the length of each diagonal.
The diagonals of a rectangle are .
8x + 2 = 5x + 11
Diagonal =
3x = 9
8x + 2
x = 3
8(3)+ 2 = 26
Since diagonals are  each is 26.

11. a. The quadrilateral has congruent diagonals and one angle of 60.
EXAMPLE #3
Determine whether the quadrilateral can be a parallelogram. If not, write impossible. Explain.
a. The quadrilateral has congruent diagonals and one angle of 60.
Impossible.
A parallelogram with congruent diagonals is a rectangle with four right angles.

12. The quadrilateral has perpendicular diagonals and four right angles.
The figure can be a parallelogram.  diagonals makes it a rhombus, 4 right angles makes it a rectangle.
If a parallelogram is both a rhombus and a rectangle then it is a square.

13. Assignment: pg 315 #1-15 odd, 16-21, 48-49, 57-58
Must include equations and explanations whenever required.