Quadrilaterals
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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 Properties-
Sides Opposite sides
are parallel
Opposite sides are congruent
Parallelogram Properties-
Angles
Opposite angles are congruent
Consecutive angles are supplementary
Parallelogram Properties-
Diagonals
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 s are
Diagonals bisect each other
RectanglesA 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 s are
congruent
Consecutive s
are supplementary
Rectangle Properties-
Diagonals
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 Properties-
Diagonals
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 s
Square - Area
A = s2
Area equals the side times the side.
Trapezoids …
Trapezoids only have one pair of parallel sides.
A Trapezoid is NOTa 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.
QuadrilateralsGeneral Quadrilateral
4 sides, 4 anglesTrapezoid
Only 1 pair of parallel sides
QuadrilateralsGeneral Quadrilateral
4 sides, 4 anglesTrapezoid
Only 1 pair of parallel sides
Parallelogram
Opposite sides are parallel and congruent
QuadrilateralsGeneral Quadrilateral
4 sides, 4 anglesTrapezoid
Only 1 pair of parallel sides
Parallelogram
Opposite sides are parallel and congruent
Rectangle
A parallelogram with 4 right angles
QuadrilateralsGeneral Quadrilateral
4 sides, 4 anglesTrapezoid
Only 1 pair of parallel sides
Parallelogram
Opposite sides are parallel and congruent
Rectangle
A parallelogram with 4 right angles
Square
A rectangle with 4 congruent sides
QuadrilateralsGeneral Quadrilateral
4 sides, 4 anglesTrapezoid
Only 1 pair of parallel sides
Parallelogram
Opposite sides are parallel and congruent
Rectangle
A parallelogram with 4 right angles
Rhombus
A parallelogram with 4 congruent sides
Square
A rectangle with 4 congruent sides
Comparison of Quadrilateral
PropertiesParallelogram 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
Diagonals Yes /No Yes /No Yes /No Yes /No Yes /No
Diagonals Yes /No 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 s
Yes /No Yes /No Yes /No Yes /No Yes /No
All s 90° Yes /No Yes /No Yes /No Yes /No Yes /No
Comparison of Quadrilateral
PropertiesParallelogram 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 Yes Yes
Diagonals No No Yes Yes No
Diagonals bisect each other
Yes Yes Yes Yes No
Diagonals bisect opposite s
No No Yes Yes No
All s 90° No Yes Yes No No
Venn Diagram of Quadrilaterals
Trapezoids
Parallelograms
Rectangles
Squares
Rhombi
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