Transcript quadrilateral

Quadrilaterals
Today’s Learning Goals
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We will learn that quadrilaterals are NOT stable
figures that keep their shape under stress.
We will understand the quadrilateral inequality – the
sum of the lengths of any three sides of a
quadrilateral is greater than the length of the fourth
side.
We will determine the sum of the interior angles for
any quadrilateral.
Definitions
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a)
A quadrilateral is a closed, four-sided 2-D figure
with straight sides that do not overlap.
Which of the following is a quadrilateral? Explain
how you know.
b)
c)
d)
e)
Good…b), d), and e) are quadrilaterals. a) is not because the
sides overlap and c) is not because one side is not straight.
Parallelograms
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A parallelogram is a quadrilateral where both pairs
of opposite sides are parallel.
Below are several examples of parallelograms.
What do you notice is true about opposite sides other
than they are parallel?
Yes…opposite sides look like they are the same
length for parallelograms.
Parallelograms
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Consider the following parallelogram:
B
C
A
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D
To prove that opposite sides have the same length,
then construct AC.
When we construct AC, two triangles are created.
How could we name the two triangles?
Nice…ABC and ADC.
Parallelograms
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Consider the highlighted angle. Since AD || BC and
AC is a transversal, which numbered angle is equal to
the highlighted angle?
B
C
3
2
4
A
1
D
Great…m(1) = m(3) because they are
alternate interior angles.
Parallelograms
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Now, consider the blue highlighted angle. Knowing
that CD || AB and AC is a transversal, which
numbered angle is equal to the blue highlighted
angle?
B
C
3
2
4
A
1
D
Great…m(4) = m(2) because they are also
alternate interior angles.
Parallelograms
Let’s break the parallelogram up into the two triangles.
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B
4
A
C
3
1
B
2
A
4
C
2
3
C
1
A
D
D
Parallelograms
From ASA, we see that ABC  ADC.
B
B
C
3
2
A 4
D
4
1
D
C 2
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A
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3
C
1
A
If ABC  ADC, then what sides are equal in the
parallelogram?
Beautiful…AB = CD and BC = AD
So, we just proved that opposite sides of a
parallelogram are equal for any parallelogram.
Other Quadrilaterals
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A rectangle is a quadrilateral with exactly four right
angles.
Which of the following are rectangles?
a)
b)
c)
d)
Yes…a, b, and c are all rectangles because they are
quadrilaterals with four right angles.
 Notice that a square is a rectangle because it satisfies
the definition of a rectangle!
Other Quadrilaterals
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A rhombus is a quadrilateral with all four sides
having the same length.
Which of the following are rhombi?
a)
b)
c)
d)
Great…b, c, and d are all rhombi because they all have four
side lengths with the same measurement.
 Notice that a square is also a rhombus because it
satisfies the definition of a rhombus!
Other Quadrilaterals
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A trapezoid is a quadrilateral with exactly one pair
of parallel sides.
Which of the following are trapezoids?
a)
b)
c)
d)
Nice…d is the only trapezoid. The other shapes all have
two pairs of parallel sides.
 Notice that trapezoids are not parallelograms while
rectangles, squares, and rhombi are parallelograms!
Quadrilateral Construction
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If you were given four different side lengths, would
you always be able to make a quadrilateral?
Okay…some people think you would be able to and
some think you might not.
Today, we are going to try to make quadrilaterals
using metal polystrips with different side lengths.
Quadrilateral Construction
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Let’s try to make a quadrilateral with our metal
polystrips with lengths of 8, 8, 8, and 8 units.
What shape is made from four 8 side lengths?
Nice…it could be a square, rectangle, rhombus,
and/or a parallelogram.
Notice how a quadrilateral is NOT rigid like the
triangle was.
How could we make the quadrilateral rigid?
Great…put a length across the diagonal to make
two triangles from the quadrilateral.
Quadrilateral Construction
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Now, let’s try to make a quadrilateral with side
lengths of 4, 4, 6, and 17 units by putting the 17
length on the bottom.
How come we could not make a quadrilateral with
side lengths of 4, 4, 6, and 17?
Yes…4 + 4 + 6 < 17 so the sides will never meet to
make a quadrilateral.
Quadrilateral Construction
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Some of you thought that a quadrilateral could be
made with any side lengths. We just saw an example
of four side lengths that did not make a quadrilateral.
Try to make more quadrilaterals with different side
lengths greater than 3 and less than 18.
Partner Work
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You have 20 minutes to work on the following
problems with your partner.
For those that finish early
Determine which set or sets of side lengths below can
make the following shapes.
i) A quadrilateral with all angles the same size.
ii) A parallelogram.
iii) A quadrilateral that is not a parallelogram.
a)
b)
c)
d)
5, 5, 8, 8
5, 5, 6, 14
8, 8, 8, 8
4, 3, 5, 14
Big Idea from Today’s Lesson
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A quadrilateral can only be made if the longest side is
shorter than the sum of the other three sides
(Quadrilateral Inequality)!
A quadrilateral is NOT rigid…more than 1
quadrilateral can be made from four given sides by
pressing on one of the corners.
The sum of the interior angles is 360° for ANY
quadrilateral.
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A rhombus is a parallelogram with four sides of equal length.
A square is a parallelogram with four sides of equal length
and four right angles.
A rectangle is a parallelogram with four right angles.
A trapezoid is a quadrilateral with one pair of parallel sides.
Homework
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Pgs. 225 – 226 (6 – 11, 14, 15)