SPECIAL QUADRILATERALS

NON-PARALLELOGRAMS

SPECIAL QUADRILATERALS

Trapeziods

Parallelograms• Rectangles• Rhombi• Squares

KitesOther Quads

Quadrilaterals

The Tricky Trapezoid

Definition:A quadrilateral with exactly one

pair of opposite sides parallel.

Special Property (Corollary)If a quadrilateral is a trapezoid,

then the pairs of base-to-base consecutive interior angles are supplementary.

• Exactly 1 pair of opposite sides parallel

• Base-to-base consecutive interior angles are supplementary

Trapezoids

1

1

2

2

Midsegments of TrapezoidsThe midsegment is the segment

connecting the midpoints of the legs of a trapezoid. • Exactly 1 pair

of opposite sides parallel

• Base-to-base consecutive interior angles are supplementary

Trapezoids

Theorem #49:The midsegment of a trapezoid is:

1) Parallel to the bases of the trapezoid

2) Length = ½ (sum of the bases)½ (b1 + b2)

• Exactly 1 pair of opposite sides parallel

• Base-to-base consecutive interior angles are supplementary

• Midsegments • Parallel to

the bases• ½ (sum of

the bases)

Trapezoids

b2

b1

Trapezoids “HOT FACTS”4 Sides –

Quadrilateral

Odd Looking Quad!

Exactly 1 pair of opposite sides parallel

Base-Base Consecutive angles supplementary

Midsegments!Parallel to the bases½ the sum of the

bases

Proving Trapezoids are QuadrilateralDude, are you serious?

Definition:If a quadrilateral has exactly 1

pair opposite sides parallel, then the quadrilateral is a trapezoid. • EXACTLY 1 pair

of opposite sides paralleltrapezoid

Trapezoids

Midsegments:If a quadrilateral has a

midsegment that is parallel to both bases and is ½ the sum of the bases, then the quadrilateral is a trapezoid.

• EXACTLY 1 pair of opposite sides paralleltrapezoid

•Midsegments parallel AND ½*(sum of the bases)

Trapezoids

b2

b1

Area of a TrapezoidTheorem #55:

Area = ½*height*(sum of the bases)

A = ½*h*(b1 + b2)

b2

b1

h

What about Special Trapezoids?You did know they exist, right?

Definition of an Isosceles Trapezoid:A trapezoid whose legs are

congruent.

Isosceles Trapezoids

• Legs congruent

Theorem #46:A trapezoid is isosceles if and only

if each pair of base angles are congruent.

Isosceles Trapezoids

• Legs congruent

• Each pair of base angles are congruent

Theorem #48:A trapezoid is isosceles if and only

if its diagonals are congruent.Isosceles Trapezoids

• Legs congruent

•Each pair of base angles are congruent

• Diagonals congruent

Isosceles Trapezoids “HOT FACTS”

4 Sides – Quadrilateral

Bottom part of an isosceles triangle!

Exactly 1 pair of opposite sides parallel (Bases)

Base-Base Consecutive angles supplementary

Midsegments!Parallel to the bases½ the sum of the

bases

Legs are congruent

Each pair of base angles are congruent

Diagonals congruent

Everything you ever wanted to know about Trapezoids and Isosceles Trapezoids…You now knowIf you did things right, you should have only used 1 sheet of paper, right?

The “Kean” Kite

Definition:A quadrilateral that has 2 pairs of

consecutive sides congruent.• 2 pairs of consecutive sides congruent

Kite

Theorem #50:A quadrilateral is a kite if and only

if its diagonals are perpendicular.•2 pairs of consecutive sides congruent

• Diagonals perpendicular

Kite

Theorem #51:A quadrilateral is a kite if and only

if it has exactly 1 pair of opposite angles congruent. •2 pairs of

consecutive sides congruent

• Diagonals perpendicular

• Exactly one pair of opposite angles congruent

Kite

Theorem #51 ½ (or #A):A quadrilateral is a kite if and only

if its long diagonal bisects the short diagonal. • 2 pairs of

consecutive sides congruent

• Diagonals perpendicular

• Exactly one pair of opposite angles congruent

• Long diagonal bisects the Short diagonal

Kite

Kites “HOT FACTS”4 Sides –

Quadrilateral

2 Isosceles triangles with same bases

2 pairs of consecutive sides are congruent

Diagonals perpendicular

Exactly 1 pair of opposite angles congruent

Long diagonal bisects the Short diagonal

Proving a Quadrilateral is a KiteWhy not just fly one!

Definition:A quadrilateral that has 2 pairs of

consecutive sides congruent.• 2 pairs of consecutive sides congruent Kite

Kite

Theorem #50:A quadrilateral is a kite if and only

if its diagonals are perpendicular.• 2 pairs of consecutive sides congruent Kite

• Diagonals perpendicular Kite

Kite

Theorem #51:A quadrilateral is a kite if and only

if it has exactly 1 pair of opposite angles congruent. • 2 pairs of

consecutive sides congruent Kite

• Diagonals perpendicular Kite

• Exactly one pair of opposite angles congruent Kite

Kite

Theorem #51 ½ (or #A):A quadrilateral is a kite if and only

if its long diagonal bisects the short diagonal. • 2 pairs of

consecutive sides congruent Kite

• Diagonals perpendicular Kite

• Exactly one pair of opposite angles congruent Kite

• Long diagonal bisects the Short diagonal Kite

Kite

Area of a KiteTheorem #56:Area = ½*product of the diagonalsA = ½*d1*d2

d1 d

2

Everything you ever wanted to know about Kites…You now knowIf you did things right, you should have only used 1 sheet of paper, right?

Parallelograms

RhombusRectangle

Square

Trapezoids Kites

Quadrilaterals