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3 Special Figures:
The Rhombus, The Rectangle, The Square
A Retrospect
Special Parallelograms
Properties, Theorems, and Conclusions
The “Forgotten” Rhombus
Definition of a RhombusA parallelogram with ALL 4 sides
congruent.
All Properties of Parallelograms Work!
• Both pairs of
opposite sides
parallel
• All Sides
Congruent!
• Both pairs of
opposite angles
congruent
• Pairs of
consecutive
angles are
supplementary
• Diagonals bisect
each other
Rhombi = Parallelograms!
Theorem #43A quadrilateral is a rhombus if and
only if its diagonals are perpendicular.
•Both pairs of
opposite sides
parallel
• All Sides
Congruent!
• Both pairs of
opposite angles
congruent
• Pairs of
consecutive
angles are
supplementary
• Diagonals
bisect each other
• Diagonals are
perpendicular
Theorem #44A quadrilateral is a rhombus if and
only if its diagonals bisect each pair of opposite angles.
•Both pairs of
opposite sides
parallel
• All Sides
Congruent!
• Both pairs of
opposite angles
congruent
• Pairs of
consecutive angles
are supplementary
• Diagonals bisect
each other
• Diagonals are
perpendicular
•Diagonals bisect
each pair of
opposite angles
11
11
22
22
NOTE: Opposite angles are already congruent!
Example #1Name pairs of parallel segments.Name pairs of congruent
segments.Name pairs of congruent angles.
•ANSWERS:
A
E
CD
B
;AB DC AD BC
;
;
AB DC AD BC
AE EC BE ED
;A C B D AEB DEC
AED BEC
DAC BAC
DCA BCA
ABD CBD
ADB CDB
Rhombus “HOT FACTS”4 Sides – Quadrilateral
Parallelogram
2 pairs of opposite sides parallel
ALL PAIRS of opposite sides congruent
2 pairs of opposite angles congruent
4 pairs of consecutive angles supplementary
Diagonals bisect each other
Diagonals perpendicular
Diagonals bisects each pair of opposite angles
Proving A Quadrilateral Is A RhombusObviously Difficult, Secretly Simple.
Step #1: Must first show the quadrilateral is a Parallelogram!Use one of the methods for
parallelograms!
• BOTH pairs of
opposite sides
congruent
parallelogram
•BOTH pairs of
opposite angles
congruent
parallelogram
• A pair of
consecutive angles
supplementary
parallelogram
•Diagonals bisect
each other
parallelogram
•Exactly 1 pair of
opposite sides
congruent and
parallel
parallelogram
Parallelograms
Step #2: Once a parallelogram, then get specific!3 ways to show a parallelogram is
a rhombus!•
Definition of a RhombusIf a quadrilateral is a
parallelogram and all 4 sides are congruent, then the quadrilateral is a rhombus.
• Quadrilateral
Parallelogram
4 congruent
sides Rhombus
Quadrilateral Parallelogram Rhombus
Theorem #45If a quadrilateral is a
parallelogram and the diagonals are perpendicular, then the quadrilateral is a rhombus.
• Quadrilateral
Parallelogram
4 congruent
sides Rhombus
• Quadrilateral
Parallelogram
Diagonals
Perpendicular
Rhombus
Quadrilateral Parallelogram Rhombus
Theorem #46If a quadrilateral is a
parallelogram and the diagonals bisect each pair of opposite angles, then the quadrilateral is a rhombus.
• Quadrilateral
Parallelogram
4 congruent
sides Rhombus
• Quadrilateral
Parallelogram
Diagonals
Perpendicular
Rhombus
• Quadrilateral
Parallelogram
Diagonals bisect
each pair of
opposite angles
Rhombus
Quadrilateral Parallelogram Rhombus
11
11
22
22
Area of a Rhombus (Method #1)Theorem #53: Area of a RhombusArea = Base * HeightA = b*h
•
h
b
Area of a Rhombus (Method #2)Theorem #57: Area of a RhombusArea = ½ * diagonal 1 * diagonal 2A = ½ * d1 * d2
•
d1
d2
Now don’t forget about a Rhombus!If you did things right, you should have only used 1 sheet of paper, right?
Properties, Theorems, and Conclusions
The “Regal” Rectangle
Definition of a RectangleA parallelogram with ALL 4
angles congruent (ALL 4 angles are right angles)
All Properties of Parallelograms Work!
• Both pairs of
opposite sides
parallel
• 2 pairs of
opposite sides
congruent
• ALL 4 angles
congruent
• Pairs of
consecutive
angles are
supplementary
• Diagonals bisect
each other
Rectangles = Parallelograms!
Theorem #47A quadrilateral is a rectangle if
and only if its diagonals are congruent.
•Both pairs of
opposite sides
parallel
• All Angles
Congruent!
• Both pairs of
opposite angles
congruent
• Pairs of
consecutive
angles are
supplementary
• Diagonals
bisect each other
• Diagonals are
congruent
Rectangle “HOT FACTS”4 Sides –
Quadrilateral
Parallelogram
2 pairs of opposite sides parallel
2 pairs of opposite sides congruent
ALL angles congruent (ALL angles are right angles)
4 pairs of consecutive angles supplementary
Diagonals bisect each other
Diagonals Congruent
Proving A Quadrilateral Is A RectangleIs it better then a Rhombus?
Step #1: Must first show the quadrilateral is a Parallelogram!Use one of the methods for
parallelograms!
• BOTH pairs of
opposite sides
congruent
parallelogram
•BOTH pairs of
opposite angles
congruent
parallelogram
• A pair of
consecutive angles
supplementary
parallelogram
•Diagonals bisect
each other
parallelogram
•Exactly 1 pair of
opposite sides
congruent and
parallel
parallelogram
Parallelograms
Step #2: Once a parallelogram, then get specific!2 ways to show a parallelogram is
a rectangle!•
Definition of a RectangleIf a quadrilateral is a
parallelogram and has all 4 angles congruent (or all 4 angles are right angles), then the quadrilateral is a rectangle.
Quadrilateral
Parallelogram
All 4 angles
congruent (all 4
angles are right
angles)
Rectangle
Quadrilateral Parallelogram Rectangle
Theorem # 48If a quadrilateral is a
parallelogram and its diagonals are congruent, then the quadrilateral is a rectangle.
Quadrilateral
Parallelogram
All 4 angles
congruent (all 4
angles are right
angles)
Rectangle
Quadrilateral
Parallelogram
Diagonals
congruent
Rectangle
Quadrilateral Parallelogram Rectangle
Area of a Rectangle
Area = Length * Width or Base * Height
A = l * w or b * h
l
w
Respect the Rectangle!If you did things right, you should have only used 1 sheet of paper, right?
Properties, Theorems, and Conclusions
The “Sassy” Square
Definition of a SquareA parallelogram that is BOTH a
Rhombus and a Rectangle!(All 4 sides congruent)(All 4 angles congruent)
All Properties of Parallelograms Work!
• Both pairs of
opposite sides
parallel
• ALL 4 sides
congruent
• ALL 4 angles
congruent
• Pairs of
consecutive
angles are
supplementary
• Diagonals
bisect each other
Square = Parallelograms
All Properties of a Rhombus Work!
All Properties of a Rectangle Work!
• Diagonals are
perpendicular
•Diagonals bisect
each pair of
opposite angles
•Diagonals are
congruent
Squares = Parallelograms, Rhombi, and Rectangles
1 11 1
1111
Square “HOT FACTS”4 Sides – QuadrilateralParallelogram2 pairs of opposite
sides parallelALL sides congruentALL angles congruent
(ALL angles are right angles)
4 pairs of consecutive angles supplementary
Diagonals bisect each other
RhombusDiagonals
perpendicularDiagonals bisect each
pair of opposite angles
RectangleDiagonals congruent
Proving A Quadrilateral Is A SquareHow hard can this be?
Step #1: Must first show the quadrilateral is a Parallelogram!Use one of the methods for
parallelograms!
• BOTH pairs of
opposite sides
congruent
parallelogram
•BOTH pairs of
opposite angles
congruent
parallelogram
• A pair of
consecutive angles
supplementary
parallelogram
•Diagonals bisect
each other
parallelogram
•Exactly 1 pair of
opposite sides
congruent and
parallel
parallelogram
Parallelograms
Step #2: Once a parallelogram, then show it is a Rhombus!Use one of the methods for
Rhombus!• Quadrilateral
Parallelogram
4 congruent
sides Rhombus
• Quadrilateral
Parallelogram
Diagonals
Perpendicular
• Quadrilateral
Parallelogram
Diagonals bisect
each pair of
opposite angles
Parallelograms Rhombus
Step #3: Once a parallelogram and a rhombus, then show it is a rectangle!Use one of the methods for
Rectangle!
Quadrilateral
Parallelogram
All 4 angles
congruent (all 4
angles are right
angles)
Rectangle
Quadrilateral
Parallelogram
Diagonals
congruent
Rectangle
Parallelograms Rhombus Rectangle
Step #4: Call your shape a square!
Quadrilateral
Parallelogra
m Rhombus
Rectangle
Square
Area of a SquarePostulate #22
Area = Side * Side or Side SquaredA = s * s
Theorem #53Area = base * heightA = b * h
sh
b
That’s a Square, folks!If you did things right, you should have only used 1 sheet of paper, right?