quadrilaterals please click the speaker symbol on the left to hear the audio that accompanies some...

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”.

Types ofTypes ofQuadrilateralsQuadrilaterals

In General …`In General …`

Quadrilaterals are Quadrilaterals are polygons that polygons that have 4 sides and have 4 sides and 4 angles.4 angles.

ParallelogramsParallelograms

A parallelogram A parallelogram has two parallel has two parallel pairs of opposite pairs of opposite sides.sides.

Parallelogram Properties- Parallelogram Properties- SidesSides

Opposite sides Opposite sides are parallelare parallel

Opposite sides Opposite sides are congruentare congruent

Parallelogram Properties- Parallelogram Properties- AnglesAngles

Opposite Opposite angles are angles are congruentcongruent

Consecutive Consecutive angles are angles are supplementarysupplementary

m m

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

Parallelogram- AreaParallelogram- Area

A = bhA = bh

Area equals the Area equals the base times the base times the height.height.

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

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.

Rectangle Properties- SidesRectangle Properties- Sides Opposite sides Opposite sides

are parallelare parallel

Opposite sides Opposite sides are congruentare congruent

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

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.

Rectangle - AreaRectangle - Area

A = bhA = bh

Area equals the Area equals the base times the base times the height.height.

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.

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.

Rhombus Properties- SidesRhombus Properties- Sides All four sides All four sides

are congruentare congruent

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

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.

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.

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.

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

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

Square - AreaSquare - Area

A = sA = s22

Area equals the Area equals the side times the side times the side.side.

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.

Isosceles TrapezoidsIsosceles Trapezoids

Legs are Legs are ≈≈

Both pairs of Both pairs of base angles are base angles are ≈≈

Diagonals are Diagonals are ≈≈

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))

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.

QuadrilateralsQuadrilateralsGeneral Quadrilateral

4 sides, 4 anglesTrapezoid

Only 1 pair of parallel sides




Parallelogram

Opposite sides are parallel and congruent




Parallelogram


Rectangle

A parallelogram with 4 right angles




Parallelogram


Rectangle

A parallelogram with 4 right angles

Square

A rectangle with 4 congruent sides




Parallelogram


Rectangle

A parallelogram with 4 right angles Rhombus

A parallelogram with 4 congruent sides

Square

A rectangle with 4 congruent sides

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


Diagonals bisect opposite angles


All angles 90°


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

Venn Diagram of QuadrilateralsVenn Diagram of Quadrilaterals

Trapezoids

Parallelograms

Rectangles

Squares

Rhombi

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pairs of opposite angles

parallelopposite sides

congruentconsecutive

pair of opposite sides

pairs of parallel sides

pairs of base angles

nonparallel sides

equal sides