Unit 6 Quadrilaterals. Lesson 6.1 Properties of Quadrilaterals.
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QuadrilateraQuadrilateralsls
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QuadrilateralQuadrilateral: : A Polygon with 4 sides A Polygon with 4 sides and 4 angles.and 4 angles.
Sides: May or may not be Sides: May or may not be congruent.congruent.
Angles: The sum of all Angles: The sum of all interior angles in a interior angles in a quadrilateral quadrilateral is 360is 360o.o.
Quad means “4” and lateral means “sides”.Quad means “4” and lateral means “sides”.
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Types ofTypes ofQuadrilateralsQuadrilaterals
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In General …`In General …`
Quadrilaterals are Quadrilaterals are polygons that polygons that have 4 sides and have 4 sides and 4 angles.4 angles.
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ParallelogramsParallelograms
A parallelogram A parallelogram has two parallel has two parallel pairs of opposite pairs of opposite sides.sides.
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Parallelogram Properties- Parallelogram Properties- SidesSides
Opposite sides Opposite sides are parallelare parallel
Opposite sides Opposite sides are congruentare congruent
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Parallelogram Properties- Parallelogram Properties- AnglesAngles
Opposite Opposite angles are angles are congruentcongruent
Consecutive Consecutive angles are angles are supplementarysupplementary
m m
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Parallelogram Properties- Parallelogram Properties- DiagonalsDiagonals
Diagonals of a Diagonals of a parallelogram parallelogram meet at the meet at the midpoints.midpoints.
Divide the Divide the parallelogram parallelogram into two into two congruent congruent trianglestriangles
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Parallelogram- AreaParallelogram- Area
A = bhA = bh
Area equals the Area equals the base times the base times the height.height.
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TheoremsTheoremsA quadrilateral is a parallelogram if any A quadrilateral is a parallelogram if any ONEONE of the of the
following is true: following is true:
Both pairs of opposite Both pairs of opposite sidessides are || are ||
Both pairs of opposite Both pairs of opposite sidessides are are ≈≈
A pair of opposite A pair of opposite sidessides is both || and is both || and ≈≈
Both pairs of opposite Both pairs of opposite anglesangles are are ≈≈
DiagonalsDiagonals bisect each other bisect each other
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RectanglesRectanglesA rectangle has: A rectangle has: two sets of two sets of
opposite sides opposite sides parallel, and parallel, and
four right angles.four right angles.
It is also a It is also a parallelogram, parallelogram, since it has two since it has two pairs of parallel pairs of parallel sides.sides.
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Rectangle Properties- SidesRectangle Properties- Sides Opposite sides Opposite sides
are parallelare parallel
Opposite sides Opposite sides are congruentare congruent
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Rectangle Properties- AnglesRectangle Properties- Angles
Four Right AnglesFour Right Angles
Opposite angles Opposite angles are congruentare congruent
Consecutive angles Consecutive angles are supplementaryare supplementary
mm
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Rectangle Properties- Rectangle Properties- DiagonalsDiagonals
Diagonals of a Diagonals of a rectangle are rectangle are congruent.congruent.
Diagonals Diagonals bisect each bisect each other.other.
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Rectangle - AreaRectangle - Area
A = bhA = bh
Area equals the Area equals the base times the base times the height.height.
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TheoremsTheorems If a parallelogram is a rectangle, then the If a parallelogram is a rectangle, then the
diagonals are congruent.diagonals are congruent.
If the diagonals of a parallelogram are If the diagonals of a parallelogram are
congruent, then the parallelogram is a congruent, then the parallelogram is a
rectangle.rectangle.
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Rhombus …Rhombus …
A rhombus is a A rhombus is a parallelogram with parallelogram with four equal sides. four equal sides.
It is not always a It is not always a rectangle because a rectangle because a rhombus rhombus does not does not have tohave to have 4 right have 4 right angles. angles.
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Rhombus Properties- SidesRhombus Properties- Sides All four sides All four sides
are congruentare congruent
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Rhombus Properties- AnglesRhombus Properties- Angles
Same as all other Same as all other parallelograms:parallelograms:
Opposite angles Opposite angles are congruentare congruent
Consecutive Consecutive angles are angles are supplementarysupplementary
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Rhombus Properties- Rhombus Properties- DiagonalsDiagonals
Diagonals of a Diagonals of a rhombus are rhombus are perpendicular.perpendicular.
Each diagonal Each diagonal bisects the bisects the opposite opposite angles.angles.
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Rhombus- AreaRhombus- Area
A = ½ dA = ½ d11 d d22
Area equals ½ Area equals ½ times the first times the first diagonal times diagonal times the second the second diagonal.diagonal.
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TheoremsTheorems
If the diagonals of a parallelogram are If the diagonals of a parallelogram are
perpendicular, then the parallelogram perpendicular, then the parallelogram
is a rhombus.is a rhombus.
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Squares …Squares …A square has: A square has: four right anglesfour right angles four congruent four congruent
sidessides
It is also a It is also a • rectangle rectangle •
parallelogramparallelogram• rhombusrhombus
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Square PropertiesSquare Properties Opposite sides are ||Opposite sides are || All sides are All sides are ≈≈ Opposite angles are Opposite angles are ≈≈
(90°)(90°) Consecutive angles are Consecutive angles are
supplementarysupplementary Diagonals are Diagonals are ≈≈ Diagonals bisect each Diagonals bisect each
other and anglesother and angles
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Square - AreaSquare - Area
A = sA = s22
Area equals the Area equals the side times the side times the side.side.
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Trapezoids …Trapezoids …
Trapezoids only Trapezoids only have have oneone pair of pair of parallel sides. parallel sides.
A Trapezoid is A Trapezoid is NOTNOT a a parallelogram. parallelogram.
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Parts of a TrapezoidParts of a Trapezoid The parallel The parallel
sides are bases.sides are bases. The nonparallel The nonparallel
sides are legs.sides are legs. Pairs of base Pairs of base
angles share the angles share the same base. same base.
M and H are a pair of base angles. A and T are a pair of base angles.
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Isosceles TrapezoidsIsosceles Trapezoids
Legs are Legs are ≈≈
Both pairs of Both pairs of base angles are base angles are ≈≈
Diagonals are Diagonals are ≈≈
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Median of a TrapezoidMedian of a Trapezoid Median is a segment Median is a segment
that joins the that joins the midpoints of the midpoints of the legs. legs.
Parallel to the basesParallel to the bases Length is ½ the sum Length is ½ the sum
of the bases of the bases median=½ (bmedian=½ (b11 + b + b22))
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Trapezoid - AreaTrapezoid - Area
A = ½ h(bA = ½ h(b11 + b + b22))
Area equals ½ Area equals ½ the height the height times the sum times the sum of the bases.of the bases.
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QuadrilateralsQuadrilateralsGeneral Quadrilateral
4 sides, 4 anglesTrapezoid
Only 1 pair of parallel sides
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QuadrilateralsQuadrilateralsGeneral Quadrilateral
4 sides, 4 anglesTrapezoid
Only 1 pair of parallel sides
Parallelogram
Opposite sides are parallel and congruent
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QuadrilateralsQuadrilateralsGeneral 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
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QuadrilateralsQuadrilateralsGeneral 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
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QuadrilateralsQuadrilateralsGeneral 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
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Comparison of Quadrilateral PropertiesComparison 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 perpendicular
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 angles
Yes /No Yes /No Yes /No Yes /No Yes /No
All angles 90°
Yes /No Yes /No Yes /No Yes /No Yes /No
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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 perpendicular
No 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
Comparison of Quadrilateral PropertiesComparison of Quadrilateral Properties
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Venn Diagram of QuadrilateralsVenn Diagram of Quadrilaterals
Trapezoids
Parallelograms
Rectangles
Squares
Rhombi