Download Lesson 7.4 Properties of Special Parallelograms

Lesson 7.4
Properties of Special
Parallelograms
Essential Question:
What are the properties of the
diagonals of rectangles, rhombuses,
and squares?
What you will learn
• Use properties of special parallelograms
• Use properties of diagonals of special
parallelograms
• Use coordinate geometry to identify special
types of parallelograms
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
Some
Quadrilaterals
Quadrilaterals are
Parallelograms.
Parallelograms All Parallelograms
are quadrilaterals.
Rhombuses
Rectangles
Squares
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Learning the relations
Quadrilaterals Some Quadrilaterals
are Rectangles.
Parallelograms All Rectangles are
Parallelograms…
Rhombuses
Rectangles
Squares
…and
quadrilaterals, too.
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Learning the relations
Some
Quadrilaterals
Quadrilaterals are
Rhombuses.
All Rhombuses are
Parallelograms…
Parallelograms
Rhombuses
…and
quadrilaterals, too.
Rectangles
Squares
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Another View
Polygons
Quadrilaterals
Rhombuses
Squares
Parallelograms
Rectangles
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Sometimes, Always, Never
Quadrilaterals
A Parallelogram
is a Rhombus.
Parallelograms
Sometimes
Rhombuses
Rectangles
Squares
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Sometimes, Always, Never
Quadrilaterals
A Square is a
Rectangle.
Parallelograms
Always
Rhombuses
Rectangles
Squares
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Sometimes, Always, Never
Quadrilaterals
A Parallelogram is
a Square.
Parallelograms
Sometimes
Rhombuses
Rectangles
Squares
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Sometimes, Always, Never
Quadrilaterals
A Rhombus is a
Rectangle.
Parallelograms
Sometimes – if
it’s a square.
Rhombuses
Rectangles
Squares
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Sometimes, Always, Never
Quadrilaterals
A Square is a
Parallelogram.
Parallelograms
Always
Rhombuses
Rectangles
Squares
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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: Proof
1. 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
1
2
5. Label 1 and 2.
6. Using CPCTC, 1  2.
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Th. 7.11: Proof (continued)
7. 1 and 2 form a
linear pair.
8. 1 + 2 = 180
1
2
9. 1 + 1 = 180
10. 21 = 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.
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Finding Angle Measures in a Rhombus
Find the measures of the numbered
angles in rhombus ABCD.
m1= 90
m2 = 61
m3 = 61
m1 + m3 + m4 = 180
90 + 61 + m4 = 180
151 + m4 = 180
m4 = 29
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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Do you know properties of a
rectangle, rhombus, and square?
• Which have congruent sides?
• Rhombus & Square
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Do you know properties of a
rectangle, rhombus, and square?
• Congruent diagonals?
• Rectangle & Square
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Do you know properties of a
rectangle, rhombus, and square?
• Which have four right angles?
• Rectangle & Square
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Do you know properties of a
rectangle, rhombus, and square?
• Which have diagonals that bisect each other?
• Rectangle & Rhombus & 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,
rectangle
the parallelogram is a ______________.
• If diagonals of a parallelogram are perpendicular,
rhombus
the parallelogram is a ___________.
• If the diagonals of a parallelogram are
perpendicular and congruent, the parallelogram is a
square
_____________.
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Figures inherit properties as we read down the
chart.
Parallelograms
Rhombuses
Rectangles
Diagonals 
Diagonals 
Squares
Diagonals  and 
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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.
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:
Q(1, 4)
P(-3, 1)
R(1, -1)
S(-3, -4)
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What shape is it?
Q(1, 4)
1. Show all sides
congruent (we know
P(-3, 1)
R(1, -1)
S(-3, -4)
To prove it is a
rhombus you could:
opposite sides are
parallel).
2. Or, show the
diagonals are
perpendicular.
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Find slopes of diagonals.
Q(1, 4)
mQS
P(-3, 1)
mPR
R(1, -1)
S(-3, -4)
4  (4) 8

 2
1  (3) 4
1  (1) 2
1



3  1 4
2
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Recall: lines are perpendicular if the product of
slopes is -1.
Q(1, 4)
P(-3, 1)
R(1, -1)
S(-3, -4)
 1
2     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
Rhombuses
Squares
Parallelograms
Rectangles
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