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Quadrilaterals
Quadrilateral:
A Polygon with 4 sides
and 4 angles.
Sides: May or may not be
congruent.
Angles: The sum of all
interior angles in a
quadrilateral is 360o.
Quad means “4” and lateral means “sides”.
Types of
Quadrilaterals
In General …`

Quadrilaterals are
polygons that
have 4 sides and
4 angles.
Parallelograms

A parallelogram
has two parallel
pairs of opposite
sides.
Parallelogram PropertiesSides


Opposite sides
are parallel
Opposite sides
are congruent
Parallelogram PropertiesAngles


Opposite angles
are congruent
Consecutive
angles are
supplementary
m
m
Parallelogram PropertiesDiagonals


Diagonals of a
parallelogram
meet at the
midpoints.
Divide the
parallelogram
into two
congruent
triangles
Parallelogram- Area
A = bh
Area equals the
base times the
height.
Theorems
A quadrilateral is a parallelogram if any
ONE of the following is true:

Both pairs of opposite sides are ||

Both pairs of opposite sides are ≈

A pair of opposite sides is both || and ≈

Both pairs of opposite angles are ≈

Diagonals bisect each other
Rectangles
A rectangle has:
 two sets of
opposite sides
parallel, and
 four right angles.
It is also a
parallelogram,
since it has two
pairs of parallel
sides.
Rectangle Properties- Sides


Opposite sides
are parallel
Opposite sides
are congruent
Rectangle Properties- Angles



Four Right Angles
Opposite angles
are congruent
Consecutive
angles are
supplementary
m
m
Rectangle PropertiesDiagonals


Diagonals of a
rectangle are
congruent.
Diagonals
bisect each
other.
Rectangle - Area
A = bh
Area equals the
base times the
height.
Theorems

If a parallelogram is a rectangle, then
the diagonals are congruent.

If the diagonals of a parallelogram are
congruent, then the parallelogram is a
rectangle.
Rhombus …


A rhombus is a
parallelogram with
four equal sides.
It is not always a
rectangle because a
rhombus does not
have to have 4 right
angles.
Rhombus Properties- Sides

All four sides
are congruent
Rhombus Properties- Angles
Same as all other
parallelograms:
 Opposite angles
are congruent

Consecutive
angles are
supplementary
Rhombus PropertiesDiagonals


Diagonals of a
rhombus are
perpendicular.
Each diagonal
bisects the
opposite
angles.
Rhombus- Area
A = ½ d1 d2
Area equals ½
times the first
diagonal times
the second
diagonal.
Theorems

If the diagonals of a parallelogram are
perpendicular, then the parallelogram
is a rhombus.
Squares …
A square has:
 four right angles
 four congruent
sides
It is also a
• rectangle
• parallelogram
• rhombus
Square Properties






Opposite sides are ||
All sides are ≈
Opposite angles are ≈
(90°)
Consecutive angles
are supplementary
Diagonals are ≈
Diagonals bisect each
other and angles
Square - Area
A = s2
Area equals the
side times the
side.
Trapezoids …


`
Trapezoids only
have one pair of
parallel sides.
A Trapezoid is NOT
a parallelogram.
Parts of a Trapezoid



The parallel
sides are bases.
The nonparallel
sides are legs.
Pairs of base
angles share the
same base.
M and  H are a pair of base angles.
 A and  T are a pair of base angles.
Isosceles Trapezoids



Legs are ≈
Both pairs of
base angles are
≈
Diagonals are ≈
Median of a Trapezoid



Median is a segment
that joins the
midpoints of the
legs.
Parallel to the bases
Length is ½ the sum
of the bases
median=½ (b1 + b2)
Trapezoid - Area
A = ½ h(b1 + b2)
Area equals ½
the height
times the sum
of the bases.
Quadrilaterals
General Quadrilateral
4 sides, 4 angles
Trapezoid
Only 1 pair of
parallel sides
Quadrilaterals
Parallelogram
Opposite sides are
parallel and congruent
General Quadrilateral
4 sides, 4 angles
Trapezoid
Only 1 pair of
parallel sides
Quadrilaterals
Parallelogram
Opposite sides are
parallel and congruent
Rectangle
A parallelogram
with 4 right angles
General Quadrilateral
4 sides, 4 angles
Trapezoid
Only 1 pair of
parallel sides
Quadrilaterals
Parallelogram
Opposite sides are
parallel and congruent
General Quadrilateral
4 sides, 4 angles
Rectangle
A parallelogram
with 4 right angles
Square
A rectangle with 4
congruent sides
Trapezoid
Only 1 pair of
parallel sides
Quadrilaterals
Parallelogram
Opposite sides are
parallel and congruent
General Quadrilateral
4 sides, 4 angles
Trapezoid
Only 1 pair of
parallel sides
Rectangle
A parallelogram
with 4 right angles
Rhombus
A parallelogram with
4 congruent sides
Square
A rectangle with 4
congruent sides
Comparison of Quadrilateral Properties
Parallelogram
Rectangle
Square
Rhombus
Isosceles
Trapezoid
Opposite
sides ≈
# of pairs
# of
pairs
# of
pairs
# of
pairs
# of
pairs
Opposite
sides ||
# of pairs
# of
pairs
# of
pairs
# of
pairs
# of
pairs
Yes /No
Yes /No
Yes /No Yes /No
Yes /No
Diagonals
Yes /No
perpendicular
Yes /No
Yes /No Yes /No
Yes /No
Diagonals
bisect each
other
Yes /No
Yes /No
Yes /No Yes /No
Yes /No
Diagonals
bisect
opposite
angles
Yes /No
Yes /No
Yes /No Yes /No
Yes /No
All angles
90°
Yes /No
Yes /No
Yes /No Yes /No
Yes /No
Diagonals ≈
Comparison of Quadrilateral Properties
Parallelogram
Rectangle
Square
Rhombus
Isosceles
Trapezoid
Opposite
sides ≈
2 pairs
2 pairs
2 pairs
2 pairs
1 pair
Opposite
sides ||
2 pairs
2 pairs
2 pairs
2 pairs
1 pair
Diagonals ≈
No
Yes
Yes
No
Yes
Diagonals
No
perpendicular
No
Yes
Yes
No
Diagonals
bisect each
other
Yes
Yes
Yes
Yes
No
Diagonals
bisect
opposite
angles
No
No
Yes
Yes
No
All angles
90°
No
Yes
Yes
No
No
Venn Diagram of Quadrilaterals
Parallelograms
Rectangles
Squares
Rhombi
Trapezoids