2.8.3 Special Parallelograms

Prove and apply properties of special parallelograms
Use properties of special parallelograms to solve problems

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

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

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

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

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

6. rhombus rhombus rhombus rhombus –a parallelogram with four congruent sides. (Plural
is either rhombi or rhombuses.)
Rhombus Properties Rhombus Properties Rhombus Properties Rhombus 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
23
4
5
6 7
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 len gth
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 len gth
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 10
x
+ =

10. square square square square –a quadrilateral with four right angles and four
congruent sides.
Note: A square has all of the properties of both bothboth botha rectangle
andandand anda rhombus:
• Diagonals are congruent
• Diagonals are perpendicular
• Diagonals bisect opposite angles (creating 45°angl es).