6.6Trapezoids and Kites Last set of quadrilateral properties.
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Transcript of 6.6Trapezoids and Kites Last set of quadrilateral properties.
![Page 1: 6.6Trapezoids and Kites Last set of quadrilateral properties.](https://reader037.fdocuments.in/reader037/viewer/2022103100/56649f215503460f94c39a97/html5/thumbnails/1.jpg)
6.6 Trapezoids and Kites
Last set of quadrilateral properties
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Terminology:
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Terminology:
TrapezoidKite
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Terminology:
Trapezoid
Quadrilateral with exactly one pair of parallel sides.
Kite
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Terminology:
Trapezoid
Quadrilateral with exactly one pair of parallel sides.
Kite Quadrilateral with two pairs of consecutive congruent sides, none of which are parallel.
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Start with the trapezoid
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Start with the trapezoid
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Start with the trapezoid
OParallel sides are called bases
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Start with the trapezoid
ONon parallel sides are called legs.
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Start with the trapezoid
OSince one pair is parallel
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Start with the trapezoid
OSince one pair is parallel
Angles on the same leg are supplementary.
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Now for the special
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Now for the special
OIsosceles trapezoid is a trapezoid whose legs are congruent.
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And now for the proof, drawing in perpendiculars
A B
C D E F
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Why is ?
A B
C D E F
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Remember,
A B
C D E F
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Why is ?
A B
C D E F
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As a result, ACE BDF by?
A B
C D E F
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C D by…
A B
C D E F
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As a result, A B by…
A B
C D E F
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Theorem 6-19: If a quadrilateral is an isosceles trapezoid, then each pair of
base ’s is .
A B
C D E F
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Make sure you can…
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Make sure you can…
OGiven one angle of an isosceles trapezoid, find the remaining 3 angles.
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Application: page 390 Problem 2
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Focusing on 1 section
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AC BD because?
A B
EC D
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C D by?
A B
EC D
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If we want to prove ’s ACD and BCD are congruent, what do they share?
A B
EC D
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ACD BCD by
A B
EC D
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AD BC by
A B
EC D
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Theorem 6-20: If a quadrilateral is an isosceles trapezoid, then its diagonals are
A B
EC D
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The return of midsegments
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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)
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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)
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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)
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In addition…
Much like triangles, the midsegment is parallel to
the sides it does not touch.
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So find its length?
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So find its length?
OAdd the bases and divide by 2.
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Working backwards
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Working backwards
OFormula:
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Working backwards
OFormula:OMidsegment =
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Plug in the length of the midsegment.
OFormula:OMidsegment =
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Plug in the length of a base.
OFormula:OMidsegment =
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Solve for the remaining base
OFormula:OMidsegment =
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Solve for the remaining base
OOrOArithmetically, multiply
the length of the midsegment by 2 and subtract the length of the given base.
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Here’s a problem I enjoy.
OGiven an isosceles trapezoid whose midsegment measures 50 cm and whose legs measures 24 mm. Find its perimeter.
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Now to kites:
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If we drew in a line of symmetry, where would it be?
T
K
E Y
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And now are there ’s?
T
K
E Y
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KEY TEY
T
K
E Y
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What new is congruent by CPCTC?
T
K
E Y
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These are called the non-vertex angles, because they connect the non congruent
sides
T
K
E Y
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What else is congruent by CPCTC
T
K
E Y
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What else is congruent by CPCTC?
T
K
E Y
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The original angles, E and Y, are the vertex angles, and we can conclude they
are bisected by the diagonal.
T
K
E Y
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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.
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Summarizing
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Summarizing
OVertex angles connect the congruent sides and are bisected by the diagonals.
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Summarizing
OVertex angles connect the congruent sides and are bisected by the diagonals.
ONon vertex angles connect the non-congruent sides and are congruent.
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One last property that becomes Theorem 6-22
T
K
E Y
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If we draw in both diagonals…
T
K
E Y
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If a quadrilateral is a kite, then its diagonals are perpendicular.
T
K
E Y
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Problem solving examples
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The family tree of quadrilateralsQuadrilateral: 4 sided polygons
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The family tree of quadrilaterals
Parallelograms TrapezoidsKites
Quadrilateral: 4 sided polygons
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The family tree of quadrilaterals
2 pairs of sides 1 pair of sides2 pairs of consecutive sides
Parallelograms TrapezoidsKites
Quadrilateral: 4 sided polygons
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Which group breaks down more?
2 pairs of sides 1 pair of sides2 pairs of consecutive sides
Parallelograms TrapezoidsKites
Quadrilateral: 4 sided polygons
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Which group breaks down more?
RectangleRhombus
2 pairs of sides 1 pair of sides2 pairs of consecutive sides
Parallelograms TrapezoidsKites
Quadrilateral: 4 sided polygons
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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
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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
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And if we combine the last 2?
Square
EquiangularQuadrilateral
EquilateralQuadrilateral
RectangleRhombus
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And if we combine the last 2?
RegularQuadrilateral
Square
EquiangularQuadrilateral
EquilateralQuadrilateral
RectangleRhombus
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Those are all the definitions
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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.
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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.
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In addition…
OYou need to determine the truth value (true/false) of a universal statement
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In addition…
OYou need to determine the truth value (true/false) of a universal statement
OAll rectangles are parallelograms.
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In addition…
OYou need to determine the truth value (true/false) of a universal statement
OAll rhombi are squares.