This document discusses two special types of parallelograms: rectangles and rhombuses. It provides the key properties of each shape: rectangles have four right angles and diagonals that are congruent, while rhombuses have four congruent sides and perpendicular diagonals that bisect opposite angles. Examples are given to demonstrate using these properties to solve problems, such as finding missing side lengths.

1. Special Parallelograms
The student is able to (I can):
• Prove and apply properties of special parallelograms.
• Use properties of special parallelograms to solve
problems.

2. rectanglerectanglerectanglerectangle – a parallelogram with four right angles.
If a parallelogram is a rectangle, then its diagonals are
congruent (“checking for square”).
F I
SH
FS IH≅

3. Because a rectangle is a parallelogram, it also “inherits” all of
the properties of a parallelogram:
• Opposite sides parallel
• Opposite sides congruent
• Opposite angles congruent (actually allallallall angles are
congruent, i.e. 90°)
• Consecutive angles supplementary
• Diagonals bisect each other (which means that all of the
“half-diagonals” are congruent)

4. Example Find each length.
1. LW
2. OL
3. OW
F O
WL
30
17

5. Example Find each length.
1. LW
LW = FO = 30
2. OL
OL = FW = 2(17) = 34
3. OW
ΔOWL is a right triangle, so
OW = 16
F O
WL
30
17
2 2 2
OW LW OL+ =
2
900 1156OW + =
2
256OW =
2 2 2
30 34OW + =

6. rhombusrhombusrhombusrhombus – a parallelogram with four congruent sides. (Plural
is either rhombi or rhombuses.)
Rhombus PropertiesRhombus PropertiesRhombus PropertiesRhombus Properties
If a parallelogram is a rhombus, then its diagonals are
perpendicular.

7. Rhombus Properties (cont.)Rhombus Properties (cont.)Rhombus Properties (cont.)Rhombus Properties (cont.)
If a parallelogram is a rhombus, then each diagonal bisects a
pair of opposite angles.
∠1 ≅ ∠2
∠3 ≅ ∠4
∠5 ≅ ∠6
∠7 ≅ ∠8
1 2 3
4
5
67
8
Since opposite angles are
also congruent:
∠1 ≅ ∠2 ≅ ∠5 ≅ ∠6
∠3 ≅ ∠4 ≅ ∠7 ≅ ∠8

8. Examples
1. What is the perimeter of a rhombus whose side length
is 7?
2. Find the value of x
3. Find the value of y
x
8
Perimeter = 40
(3y+11)°
(13y–9)°
10

9. Examples
1. What is the perimeter of a rhombus whose side length
is 7?
4(7) = 28
2. Find the value of x
The side = 10
x = 6
3. Find the value of y
13y – 9 = 3y + 11
10y = 20
y = 2
x
8
Perimeter = 40
(3y+11)°
(13y–9)°
10
2 2 2
8 10x + =

10. squaresquaresquaresquare – a quadrilateral with four right angles and four
congruent sides.
Note: A square has all of the properties of bothbothbothboth a rectangle
andandandand a rhombus:
• Diagonals are congruent
• Diagonals are perpendicular
• Diagonals bisect opposite angles (creating 45° angles).