Properties of Special Parallelograms

Lesson 7.4

Essential Question: What are the properties of the diagonals of rectangles, rhombuses, and squares?

• Use properties of special parallelograms

• Use properties of diagonals of special parallelograms

• Use coordinate geometry to identify special types of parallelograms

What you will learn

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Parallelogram

•A quadrilateral with both pairs of opposite sides parallel.

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Rhombus

• Parallelogram with four congruent sides.

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Rectangle

•A parallelogram with four right angles.

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Square

•A parallelogram with four congruent sides and four right angles.

•A rectangle with four congruent sides.

•A rhombus with four right angles.

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How these relate

Quadrilaterals

Parallelograms

Rhombuses Rectangles

Squares

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Learning the relations

Quadrilaterals

Parallelograms

Rhombuses Rectangles

Squares

Some

Quadrilaterals are

Parallelograms.

All Parallelograms

are quadrilaterals.

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Learning the relations

Quadrilaterals

Parallelograms

Rhombuses Rectangles

Squares

Some Quadrilaterals

are Rectangles.

All Rectangles are

Parallelograms…

…and

quadrilaterals, too.

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Learning the relations

Quadrilaterals

Parallelograms

Rhombuses Rectangles

Squares

Some

Quadrilaterals are

Rhombuses.

All Rhombuses are

Parallelograms…

…and

quadrilaterals, too.

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Another View

Polygons

Quadrilaterals

Parallelograms

Rhombuses Rectangles

Sq

ua

res

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Sometimes, Always, Never

Quadrilaterals

Parallelograms

Rhombuses Rectangles

Squares

A Parallelogram

is a Rhombus.

Sometimes

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Sometimes, Always, Never

Quadrilaterals

Parallelograms

Rhombuses Rectangles

Squares

A Square is a

Rectangle.

Always

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Sometimes, Always, Never

Quadrilaterals

Parallelograms

Rhombuses Rectangles

Squares

A Parallelogram is

a Square.

Sometimes

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Sometimes, Always, Never

Quadrilaterals

Parallelograms

Rhombuses Rectangles

Squares

A Rhombus is a

Rectangle.

Sometimes – if

it’s a square.

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Sometimes, Always, Never

Quadrilaterals

Parallelograms

Rhombuses Rectangles

Squares

A Square is a

Parallelogram.

Always

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This chart should be memorized.

Quadrilaterals

Parallelograms

Rhombuses Rectangles

Squares

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Figures “inherit” properties as we read down the chart.

Quadrilaterals

Parallelograms

Rhombuses Rectangles

Squares

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Example

• True or False?

• The diagonals of a rectangle bisect each other.

• What we know: the diagonals of a parallelogram bisect each other.

• A rectangle is a parallelogram.

• The statement is TRUE.

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Another Example

• True or False?

• Opposite sides of a square are parallel.

• What we know: All squares are parallelograms.

• Opposite sides of a parallelogram are parallel.

• Therefore, opposite sides of a square are parallel. The statement is TRUE.

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Rhombus Properties

• The diagonals of a rhombus are perpendicular.

• The converse is true:

• If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

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Th. 7.11: The diagonals of a rhombus are perpendicular.

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Th. 7.11: Proof1. By definition, four

sides are congruent.

2. Draw the

diagonals.

3. The diagonals of

a parallelogram

bisect each other.

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Th. 7.11: Proof (continued)

4. Four triangles are

congruent. Why?

SSS

5. Label 1 and 2.

1

2

6. Using CPCTC, 1 2.

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Th. 7.11: Proof (continued)

1

2

7. 1 and 2 form a linear pair.

8. 1 + 2 = 180

9. 1 + 1 = 180

10. 21 = 180

11. 1 = 90

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Th. 7.11: Proof (continued)

1

2

12. 1 is a right angle and the diagonals are

perpendicular.

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Theorem 7.12

• The diagonals of a rhombus bisect the angles.

Find the measures of the numbered angles in rhombus ABCD.

m1= 90

m2 = 61

m3 = 61

m1 + m3 + m4 = 180

90 + 61 + m4 = 180

151 + m4 = 180

m4 = 29

Finding Angle Measures in a Rhombus

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Theorem 7.13

• The diagonals of a rectangle are congruent.

The converse is: if the diagonals of a

parallelogram are congruent, then

the parallelogram is a rectangle.

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If the diagonals of

a cabinet are

equal, then I know

it’s “square”.

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• Which have congruent sides?

• Rhombus & Square

Do you know properties of a

rectangle, rhombus, and square?

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• Congruent diagonals?

• Rectangle & Square

Do you know properties of a

rectangle, rhombus, and square?

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• Which have four right angles?

• Rectangle & Square

Do you know properties of a

rectangle, rhombus, and square?

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• Which have diagonals that bisect each other?

• Rectangle & Rhombus & Square

Do you know properties of a

rectangle, rhombus, and square?

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True or False?

• The Diagonals of a square are perpendicular.

• TRUE!

• Reason: the diagonals of a rhombus are perpendicular, and a square is a rhombus.

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Diagonals Tell Us a Lot.

• If the diagonals of a parallelogram are congruent, the parallelogram is a ______________.

• If diagonals of a parallelogram are perpendicular, the parallelogram is a ___________.

• If the diagonals of a parallelogram are perpendicular and congruent, the parallelogram is a _____________.

rectangle

rhombus

square

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Figures inherit properties as we read down the chart.

Parallelograms

Rhombuses Rectangles

Squares

Diagonals Diagonals

Diagonals and

Warm Up

2/5 Warm Up

1. A ___________________ and a ___________________ are parallelograms with four

congruent sides.

2. A ___________________ and a ______________________ are parallelograms with four right

angles.

3. A _______________________ is a parallelogram with four congruent sides and four right

angles.

4. A quadrilateral is a rhombus if and only if it has four ___________________

____________________.

5. A quadrilateral is a rectangle “iff” is has four ______________________

____________________.

6. A quadrilateral is a square “iff” it is a _______________________ and a

______________________.

7. A parallelogram is a _____________________ or a ____________________ “iff” its

diagonals are ┴ .

8. A parallelogram is a __________________ or a _______________ “iff” its diagonals are

angle bisectors.

9. A parallelogram is a ___________________ or a __________________ “iff” its diagonals are

congruent.

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Coordinate Geometry

• Given a parallelogram is formed by the points

• P(-3, 1)

• Q(1, 4)

• R(1, -1)

• S(-3, -4)

• Graph these points.

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This is the graph:

P(-3, 1)

Q(1, 4)

R(1, -1)

S(-3, -4)

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What shape is it?

P(-3, 1)

Q(1, 4)

R(1, -1)

S(-3, -4)

To prove it is a rhombus you could:

1.Show all sides congruent (we know opposite sides are parallel).

2.Or, show the diagonals are perpendicular.

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Find slopes of diagonals.

P(-3, 1)

Q(1, 4)

R(1, -1)

S(-3, -4)

4 ( 4) 8

1 ( 3) 4

1 ( 1) 2

3 1 4

2

1

2

QS

PR

m

m

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Recall: lines are perpendicular if the product of slopes is -1.

P(-3, 1)

Q(1, 4)

R(1, -1)

S(-3, -4)

12 1

2

Diagonals are

perpendicular

so the

parallelogram

is a rhombus.

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Whadaya know?

• Can you tell the difference between a rectangle and a rhombus?

• A rhombus and a square?

• A rectangle and a square?

• Can you prove what kind of parallelogram one is from given information?

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Another View

Polygons

Quadrilaterals

Parallelograms

Rhombuses Rectangles

Sq

ua

res