6.6Trapezoids and Kites Last set of quadrilateral properties.

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


OParallel sides are called bases


ONon parallel sides are called legs.


OSince one pair is parallel


OSince one pair is parallel

Angles on the same leg are supplementary.

Now for the special

Now for the special

OIsosceles 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


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)


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.


Plug in the length of a base.


Solve for the remaining base



OOr

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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 Y

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

Summarizing

Summarizing

OVertex angles connect the congruent sides and are bisected by the diagonals.

Summarizing

OVertex 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



Which group breaks down more?





RectangleRhombus





EquiangularQuadrilateral

EquilateralQuadrilateral

RectangleRhombus




And if we combine the last 2?



RectangleRhombus





Square



RectangleRhombus


RegularQuadrilateral

Square



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…


OAll rectangles are parallelograms.

In addition…


OAll rhombi are squares.

6.6Trapezoids and Kites Last set of quadrilateral properties.

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Transcript of 6.6Trapezoids and Kites Last set of quadrilateral properties.