Transcript Quadrilaterals

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Information
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© Boardworks 2012
What is a quadrilateral?
A quadrilateral is a four-sided polygon.
A quadrilateral is the polygon
with the fewest number of sides
that allows for it to be a concave
polygon (where one of the
interior angles is greater than
180 degrees).
Why is it impossible to have a concave triangle?
Since the interior angles of a triangle have to sum to 180°, there can
not be an interior angle greater than 180 degrees.
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Quadrilateral angle sum
Polygon angle sum theorem:
The sum of the measures of the interior angles of
a convex n-sided polygon is (n – 2) 180°
Using this theorem, we can see that the sum of the
measures of a quadrilateral is: 2 × 180° = 360°.
The angle sum theorem of a
quadrilateral can also be explained
in a diagram. A quadrilateral can be
divided into two triangles by drawing
a line between two opposite
C
vertices. Each of these triangles
has an angle sum of 180 degrees.
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B
A
D
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Quadrilateral angle sum
Prove that the interior angle sum of
quadrilateral ABCD is 360°.
A
1
3
given: quadrilateral ABCD,
C
split by BC
hypothesis: mA + mB + mC + mD = 360º
triangle angle
sum theorem:
angle addition:
2 4
B
D
mA + m1 + m2 = 180°
and mD + m3 + m4 = 180º
m1 + m3 = mC and m2 + m4 = mB
angle substitution:
mA + mB + mC + mD
= mA + (m2 + m4) + (m1 + m3) + mD
group by triangles:
(mA + m1 + m2) + (mD + m3 + m4)
triangle angle
sum theorem:
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mA + mB + mC + mD = 180º + 180º = 360º
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Parallelograms
In parallelogram ABCD we know that AD || BC and AB || CD.
Diagonal BD divides the parallelogram into two triangles.
A
B
D
C
alternate interior
angles are congruent:
reflexive property:
ASA property:
corresponding parts of
congruent triangles are
congruent (CPCTC):
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Use parallelogram ABCD to prove that
opposite sides and opposite angles in
any parallelogram are congruent.
ADB ≅ DBC and ABD ≅ BDC
BD ≅ BD
△ABD ≅ △BCD
AD ≅ CB and AB ≅ CD
Also BAD ≅ BCD
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Properties of parallelograms
A bisector is a line that goes through the midpoint of a
segment and divides the line into two congruent parts.
Given parallelogram ABCD, with diagonals AC and BD
intersecting at E, prove that AE ≅ CE and BE ≅ DE (that
the diagonals of the parallelogram bisect each other)
A
B
E
D
C
congruence of alternate
interior angles:
congruence of opposite sides:
ASA property:
corresponding parts of congruent
triangles are congruent (CPCTC):
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ABD ≅ BDC
and CAB ≅ ACD
AB ≅ CD
△ABE ≅ △CDE
BE ≅ DE
and AE ≅ CE
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Proving that ABCD is a parallelogram
If ABCD is a quadrilateral, then how can we prove that it
is also a parallelogram?
We must prove that both pairs of opposite sides are parallel.
How can we prove that lines are parallel in a quadrilateral?
To prove that lines are parallel,
we must prove one of the following:
1) Alternate interior angles are congruent.
2) Corresponding angles are congruent.
3) Same side interior angles are supplementary.
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Proving that ABCD is a parallelogram
If the opposite angles of a quadrilateral are congruent,
then the quadrilateral is a parallelogram.
Given quadrilateral ABCD with A ≅ C and
B ≅ D, prove that ABCD is a parallelogram.
A
b
a
B
a
b
C
D
polygon angle
sum theorem:
A≅C and B≅D:
group like terms:
divide by 2:
a and b are supplementary:
converse of the alternate
interior angle theorem:
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mA + mB + mC + mD = 360°
a + a + b + b = 360°
2a + 2b = 360°
a + b = 180°
A and D are supplementary
D and C are supplementary
AB || CD and AD || BC 
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Proof of a parallelogram
With a partner, use congruent triangles to prove that if the
opposite sides of a quadrilateral are congruent, then
it is a parallelogram.
A
B
given:
hypothesis:
D
C
reflexive property:
SSS congruence postulate:
CPCTC:
converse of the alternate
interior angle theorem:
given AB || CD and AD || BC:
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quadrilateral ABCD with
AB ≅ CD and AD ≅ BC
ABCD is a parallelogram
BD ≅ BD
△ABD ≅ △CBD
ABD ≅ BDC
and CBD ≅ ADB
Since ABD ≅ BDC, AB || CD
Since CBD ≅ ADB, AD || BC
ABCD is a parallelogram.
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Rectangles and parallelograms
Prove that if the diagonals of a parallelogram are
congruent, then the parallelogram must be a rectangle.
A
B
given:
hypothesis:
D
C
congruence of opposite
sides of a parallelogram:
AD ≅ BC and AB ≅ CD
SSS congruence postulate:
△ADC ≅ △BCD
CPCTC:
since AD || BC and they are
same side interior angles:
ADC ≅ BCD
ADC and BCD are
congruent and supplementary:
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parallelogram ABCD
where AC ≅ BD
ABCD is a rectangle
ADC and BCD
are supplementary
mADC = mBCD = 90º 
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