6.6trapezoids and kites

6.6 Trapezoids and Kites

Last set of quadrilateral properties

Terminology:

Terminology:TrapezoidKite

Terminology:Trapezoid

Quadrilateral with exactly one pair of parallel sides.

Kite

Terminology:Trapezoid

Quadrilateral with exactly one pair of parallel sides.

Kite Quadrilateral with two pairs of consecutive congruent sides, none of which are parallel.

Start with the trapezoid

Start with the trapezoid

Start with the trapezoidOParallel sides are called bases

Start with the trapezoidONon parallel sides are called legs.

Start with the trapezoidOSince one pair is parallel

Start with the trapezoidOSince one pair is parallel

Angles on the same leg are supplementary.

Now for the special

Now for the specialOIsosceles trapezoid is a trapezoid whose legs are congruent.

And now for the proof, drawing in perpendiculars

A B

C D E F

Why is ?

A B

C D E F

Remember,

A B

C D E F

Why is ?

A B

C D E F

As a result, ACE BDF by?

A B

C D E F

C D by…

A B

C D E F

As a result, A B by…

A B

C D E F

Theorem 6-19: If a quadrilateral is an isosceles trapezoid, then each pair of base

’s is .

A B

C D E F

Make sure you can…

Make sure you can…OGiven one angle of an isosceles trapezoid, find the remaining 3 angles.

Application: page 390 Problem 2

Focusing on 1 section

AC BD because?

A B

EC D

C D by?

A B

EC D

If we want to prove ’s ACD and BCD are congruent, what do they share?

A B

EC D

ACD BCD by A B

EC D

AD BC by A B

EC D

Theorem 6-20: If a quadrilateral is an isosceles trapezoid, then its diagonals are

A B

EC D

The return of midsegments

The return of midsegments

A midsegment of a trapezoid connects the midpoints of the legs (non parallel sides) and is the mean value of the 2 bases

(parallel sides)

The return of midsegments

A midsegment of a trapezoid connects the midpoints of the legs

(non parallel sides) and is the mean value of the 2 bases (parallel sides)

In addition…

A midsegment of a trapezoid connects the midpoints of the legs

(non parallel sides) and is the mean value of the 2 bases (parallel sides)

In addition…

Much like triangles, the midsegment is parallel to

the sides it does not touch.

So find its length?

So find its length?OAdd the bases and divide by 2.

Working backwards

Working backwards

OFormula:

Working backwards

OFormula:OMidsegment =

Plug in the length of the midsegment.

OFormula:OMidsegment =

Plug in the length of a base.

OFormula:OMidsegment =

Solve for the remaining base

OFormula:OMidsegment =

Solve for the remaining base

OOr

Solve for the remaining base

OOrOArithmetically, multiply the length of the midsegment by 2 and subtract the length of the given base.

Here’s a problem I enjoy.

OGiven an isosceles trapezoid whose midsegment measures 50 cm and whose legs measures 24 mm. Find its perimeter.

Now to kites:

If we drew in a line of symmetry, where would it be?

T

K

E Y

And now are there ’s?

T

K

E Y

KEY TEY

T

K

E Y

What new is congruent by CPCTC?

T

K

E Y

These are called the non-vertex angles, because they connect the non congruent

sides

T

K

E Y

What else is congruent by CPCTC

T

K

E Y

What else is congruent by CPCTC?

T

K

E Y

The original angles, E and Y, are the vertex angles, and we can conclude they are

bisected by the diagonal.

T

K

E Y

The original angles, E and Y, are the vertex angles, and we can conclude they are

bisected by the diagonal.

T

K

E YThe vertex angles of a kite are the common endpoints of the congruent sides.

Summarizing

SummarizingOVertex angles connect the

congruent sides and are bisected by the diagonals.

SummarizingOVertex angles connect the

congruent sides and are bisected by the diagonals.

ONon vertex angles connect the non-congruent sides and are congruent.

One last property that becomes Theorem 6-22

T

K

E Y

If we draw in both diagonals…

T

K

E Y

If a quadrilateral is a kite, then its diagonals are perpendicular.

T

K

E Y

Problem solving examples

The family tree of quadrilateralsQuadrilateral: 4 sided polygons

The family tree of quadrilaterals

Parallelograms TrapezoidsKites

Quadrilateral: 4 sided polygons

The family tree of quadrilaterals

2 pairs of sides 1 pair of sides2 pairs of consecutive sides

Parallelograms TrapezoidsKites

Quadrilateral: 4 sided polygons

Which group breaks down more?

2 pairs of sides 1 pair of sides2 pairs of consecutive sides

Parallelograms TrapezoidsKites

Quadrilateral: 4 sided polygons

Which group breaks down more?

RectangleRhombus

2 pairs of sides 1 pair of sides2 pairs of consecutive sides

Parallelograms TrapezoidsKites

Quadrilateral: 4 sided polygons

Which group breaks down more?

EquiangularQuadrilateralEquilateral

Quadrilateral

RectangleRhombus

2 pairs of sides 1 pair of sides2 pairs of consecutive sides

Parallelograms TrapezoidsKites

Quadrilateral: 4 sided polygons

And if we combine the last 2?

EquiangularQuadrilateralEquilateral

Quadrilateral

RectangleRhombus

2 pairs of sides 1 pair of sides2 pairs of consecutive sides

Parallelograms TrapezoidsKites

Quadrilateral: 4 sided polygons

And if we combine the last 2?

Square

EquiangularQuadrilateralEquilateral

Quadrilateral

RectangleRhombus

And if we combine the last 2?

RegularQuadrilateral

Square

EquiangularQuadrilateralEquilateral

Quadrilateral

RectangleRhombus

Those are all the definitions

Those are all the definitions

OYou need to remember all the properties, especially the ones that work for parallelograms, since they also work for a rhombus, rectangle, and square.

In addition…OYou need to remember all the properties, especially the ones that work for parallelograms, since they also work for a rhombus, rectangle, and square.

In addition…OYou need to determine the truth value (true/false) of a universal statement

In addition…OYou need to determine the truth value (true/false) of a universal statement

OAll rectangles are parallelograms.

In addition…OYou need to determine the truth value (true/false) of a universal statement

OAll rhombi are squares.