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 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.
Carpenters use this property when they “check 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 9?
2. Find the value of x
3. Find the value of y
x
8
Perimeter = 40
(3y+11)°
(13y–9)°

9. Examples
1. What is the perimeter of a rhombus whose side length
is 9?
4(9) = 36
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).