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• 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
February 4, 2016 7
Square
•A parallelogram with four congruent sides and four right angles.
•A rectangle with four congruent sides.
•A rhombus with four right angles.
February 4, 2016 9
Learning the relations
Quadrilaterals
Parallelograms
Rhombuses Rectangles
Squares
Some
Quadrilaterals are
Parallelograms.
All Parallelograms
are quadrilaterals.
February 4, 2016 10
Learning the relations
Quadrilaterals
Parallelograms
Rhombuses Rectangles
Squares
Some Quadrilaterals
are Rectangles.
All Rectangles are
Parallelograms…
…and
quadrilaterals, too.
February 4, 2016 11
Learning the relations
Quadrilaterals
Parallelograms
Rhombuses Rectangles
Squares
Some
Quadrilaterals are
Rhombuses.
All Rhombuses are
Parallelograms…
…and
quadrilaterals, too.
February 4, 2016 12
Another View
Polygons
Quadrilaterals
Parallelograms
Rhombuses Rectangles
Sq
ua
res
February 4, 2016 13
Sometimes, Always, Never
Quadrilaterals
Parallelograms
Rhombuses Rectangles
Squares
A Parallelogram
is a Rhombus.
Sometimes
February 4, 2016 14
Sometimes, Always, Never
Quadrilaterals
Parallelograms
Rhombuses Rectangles
Squares
A Square is a
Rectangle.
Always
February 4, 2016 15
Sometimes, Always, Never
Quadrilaterals
Parallelograms
Rhombuses Rectangles
Squares
A Parallelogram is
a Square.
Sometimes
February 4, 2016 16
Sometimes, Always, Never
Quadrilaterals
Parallelograms
Rhombuses Rectangles
Squares
A Rhombus is a
Rectangle.
Sometimes – if
it’s a square.
February 4, 2016 17
Sometimes, Always, Never
Quadrilaterals
Parallelograms
Rhombuses Rectangles
Squares
A Square is a
Parallelogram.
Always
February 4, 2016 18
This chart should be memorized.
Quadrilaterals
Parallelograms
Rhombuses Rectangles
Squares
February 4, 2016 19
Figures “inherit” properties as we read down the chart.
Quadrilaterals
Parallelograms
Rhombuses Rectangles
Squares
February 4, 2016 20
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.
February 4, 2016 21
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.
February 4, 2016 22
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.
February 4, 2016 24
Th. 7.11: Proof1. By definition, four
sides are congruent.
2. Draw the
diagonals.
3. The diagonals of
a parallelogram
bisect each other.
February 4, 2016 25
Th. 7.11: Proof (continued)
4. Four triangles are
congruent. Why?
SSS
5. Label 1 and 2.
1
2
6. Using CPCTC, 1 2.
February 4, 2016 26
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
February 4, 2016 27
Th. 7.11: Proof (continued)
1
2
12. 1 is a right angle and the diagonals are
perpendicular.
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
February 4, 2016 30
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.
February 4, 2016 32
• Which have congruent sides?
• Rhombus & Square
Do you know properties of a
rectangle, rhombus, and square?
February 4, 2016 33
• Congruent diagonals?
• Rectangle & Square
Do you know properties of a
rectangle, rhombus, and square?
February 4, 2016 34
• Which have four right angles?
• Rectangle & Square
Do you know properties of a
rectangle, rhombus, and square?
February 4, 2016 35
• Which have diagonals that bisect each other?
• Rectangle & Rhombus & Square
Do you know properties of a
rectangle, rhombus, and square?
February 4, 2016 36
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.
February 4, 2016 37
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
February 4, 2016 38
Figures inherit properties as we read down the chart.
Parallelograms
Rhombuses Rectangles
Squares
Diagonals Diagonals
Diagonals and
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.
February 4, 2016 41
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.
February 4, 2016 43
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.
February 4, 2016 44
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
February 4, 2016 45
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.
February 4, 2016 46
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?