1. Rhombuses, Rectangles, and Squares
Chapter 6 Section 6.4
Rhombuses, Rectangles, and Squares

2. | | A rhombus is a parallelogram with four congruent sides.
PROPERTIES OF SPEC IAL PARALLELOGRAMS
A rhombus is a parallelogram
with four congruent sides.
A rectangle is a parallelogram
with four right angles. | |
A square is a parallelogram with four congruent sides
and four right angles.

3. Always Sometimes Sometimes Always Always Always
PROPERTIES OF SPEC IAL PARALLELOGRAMS
Always
Sometimes
Sometimes
Always
Always
Always

4. mECD = 32 mADC = 2mBDC mADC = 86 = mCBA mDAB = 2mEAB
PROPERTIES OF SPEC IAL PARALLELOGRAMS
mECD = 32
mADC = 2mBDC
mADC = 86 = mCBA
mDAB = 2mEAB
mDAB = 114
mDAB + mADC = 180
114 + mADC = 180
mADC = 66

5. Diagonals are  3x – 15 = 90 3x = 105 x = 35 5x – 8 = 3x + 24
PROPERTIES OF SPEC IAL PARALLELOGRAMS
Diagonals are 
3x – 15 = 90
3x = 105
x = 35
Alt. Int Angle Thm
mADE = m CBE
5x – 8 = 3x + 24
2x – 8 = 24
2x = 32
x = 16

6. Consecutive ’s Supplementary
PROPERTIES OF SPEC IAL PARALLELOGRAMS
Consecutive ’s Supplementary
4x x + 10 = 180
6x +24 = 180
6x = 156
x = 26

7. Using Properties of Special Parallelograms
COROLLARIES ABOUT SPECIAL QUADRILATERALS
RHOMBUS COROLLARY
A quadrilateral is a rhombus if and only if it has four congruent sides.
RECTANGLE COROLLARY
A quadrilateral is a rectangle if and only if it has four right angles.
SQUARE COROLLARY
A quadrilateral is a square if and only if it is a rhombus and a rectangle.
You can use these corollaries to prove that a quadrilateral is a rhombus, rectangle, or square without proving first that the quadrilateral is a parallelogram.

8. If the Sides are , then it is a rectangle
Proving a Parallelogram is a Special Kind
If the Sides are , then it is a rectangle
P(-2, 3)
S(2, 3)
Q(-2, -4)
R(2, -4)
And
A parallelogram with 4 right angles is a Rectangle

9. If the Sides are , then it is a rhombus
Proving a Parallelogram is a Special Kind
If the Sides are , then it is a rhombus
Q(3, 6)
R(-1, -1)
P(7, -1)
S(3, -8)

10. Since PQRS is a parallelogram opposite sides are congruent
Proving a Parallelogram is a Special Kind
Since PQRS is a parallelogram opposite sides are congruent
Q(3, 6)
R(-1, -1)
S(3, -8)
All 4 Sides 
P(7, -1)
A parallelogram with 4  sides is a Rhombus

11. If diagonals are , then it is a rectangle
Proving a Parallelogram is a Special Kind
If diagonals are , then it is a rectangle
Q(3, 7)
R(6, 4)
P(-4, 0)
S(-1, -3)

12. If diagonals are , then it is a rectangle
Proving a Parallelogram is a Special Kind
If diagonals are , then it is a rectangle
R(6, 4)
P(-4, 0)
Q(3, 7)
S(-1, -3)
PQRS is a rectangle
#16

13. Need to show: All Sides  OR 2. Diagonals  Def.  ’s
Proving a Parallelogram is a Special Kind
Need to show:
All Sides  OR
2. Diagonals 
Def.  ’s

14. HW #71 Pg , 66-68, Even, 83   

15. Diagonals Bisect Each Other
Opp. Sides 
Parallelograms
Opp. ’s 
Opp. Sides ||
Cons.  Supplementary
Diagonals Bisect Each Other
Rhombus
Squares
Rectangles
4  Sides
Diagonals are 
4 Right ’s
Diagonals Bisect the Opp ’s
Diagonals are 

16. Cons. ’s Supplementary
Opp. Sides ||
Opp. Sides 
Opp. ’s 
Parallelograms
Quadrilateral
Cons. ’s Supplementary
Diagonals Bisect Each Other
One Pair Opp Sides  and ||

17. 4  Sides
Rhombus
Quadrilateral
Rectangle
4 Right ’s
Square

18. Both Diagonals Bisect each Opposite 
Parallelogram
Diagonals 
Rhombus
Both Diagonals Bisect each Opposite 
2 Consecuitive Sides 
Square
Rectangle
1 Right 