6.6trapezoids and kites last set of quadrilateral properties
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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 trapezoid
OParallel sides are called bases
Start with the trapezoid
ONon parallel sides are called legs.
Start with the trapezoid
OSince one pair is parallel
Start with the trapezoid
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
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
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
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?
EquiangularQuadrilateral
EquilateralQuadrilateral
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?
EquiangularQuadrilateral
EquilateralQuadrilateral
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
EquiangularQuadrilateral
EquilateralQuadrilateral
RectangleRhombus
And if we combine the last 2?
RegularQuadrilateral
Square
EquiangularQuadrilateral
EquilateralQuadrilateral
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.