Download PP--Special Parallelograms

PROPERTIES OF SPECIAL
PARALLELOGRAMS
Geometry CP (Holt 6-4)
K.Santos
SPECIAL PARALLELOGRAMS
Special Parallelograms:
Rectangle—a quadrilateral with
4 right angles
Rhombus—a quadrilateral with
4 congruent sides
Square—a quadrilateral with 4 right
angles & 4 congruent sides
THEOREM 6-4-1
If a quadrilateral is a rectangle, then it is a
parallelogram.
A
B
C
ABCD is a parallelogram
D
THEOREM 6-4-2
If a parallelogram is a rectangle (square) then its
diagonals are congruent.
A
B
D
Given: ABCD is a rectangle
Then: 𝐴𝐶 ≅ 𝐵𝐷
C
THEOREM 6-4-3
If a quadrilateral is a rhombus, then it is a
parallelogram.
A
B
D
ABCD is a parallelogram
C
THEOREM 6-4-4
If a parallelogram is a rhombus (or square), then
its diagonals are perpendicular
A
B
D
Given: ABCD is a rhombus
Then: 𝐴𝐶 ⟘ 𝐵𝐷
C
THEOREM 6-4-5
If a parallelogram is a rhombus (square) then each
diagonal bisects a pair of opposite angles.
A
D
B
C
Given: ABCD is a rhombus
Then: <DAB and <DCB are bisected (by 𝐴𝐶)
<ABC and <ADC are bisected (by 𝐵𝐷)
EXAMPLE: RHOMBUS
Find the missing angles:
40°
Diagonals are perpendicular
Diagonals bisect the angles
EXAMPLE: RECTANGLE
40°
Watch for isosceles triangles
EXAMPLE: SQUARE
Diagonals perpendicular
Diagonals bisect right angles
EXAMPLE:
One diagonal of a rectangle has length 8x + 2. The
other diagonal has length 5x + 11. Find the length of
each diagonal.
Diagonals of a rectangle are congruent
8x + 2 = 5x + 11
3x + 2 = 11
3x = 9
x=3
8x + 2 = 8(3) + 2 = 26
5x + 11 = 5(3) + 11 = 26
Both diagonals measure 26.
EXAMPLE:
S
T
RSTV is a rhombus.
Find each measure.
ST = 4x + 7 and SR = 9x -13.
R
All sides congruent
9x – 13= 4x + 7
5x -13 = 7
5x = 20
x=4
ST = 4(4) + 7 = 23
SR = 9(4) – 13 = 23
V
EXAMPLE:
S
RSTV is a rhombus.
m<TSW = y+2 and m<SWT = 2y+10
Find y and m<WSR.
R
Diagonals of rhombus are perpendicular
m<SWT = 90°
2y + 10 = 90
2y = 80
y = 40
Diagonal bisected the angle
m<WSR = y + 2 = 40 + 2 = 42°
T
W
V