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?