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.

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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