Download Definition: Parallelogram A parallelogram is a quadrilateral in which

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Transcript 
Definition: Parallelogram
A parallelogram is a quadrilateral in
which both pairs of opposite sides
are parallel.
A
D
B
C
A
D
B
C
If both pairs of opposite sides of a quadrilateral
are parallel, the quadrilateral is a parallelogram.
(reverse of the definition)
A
D
B
C
Here is a quadrilateral with both pairs of
opposite sides congruent.
A
D
B
C
A
D
B
C
Triangle ADC is congruent to triangle CBD by
SSS.
A
D
B
C
The marked pairs of angles are congruent by
CPCTC.
A
D
B
C
The pairs of opposite sides are parallel by
alternate interior angles.
A
D
B
C
If both pairs of opposite sides are congruent, a
quadrilateral is a parallelogram.
A
D
B
C
This quadrilateral has one pair of opposite
sides parallel and congruent.
A
D
B
C
A
D
B
C
The two angles marked in green are congruent
by alternate interior angles.
A
D
B
C
Triangle ADC is congruent to triangle CBD by
SAS.
A
D
B
C
The remaining pairs of corresponding angles
and sides are congruent by CPCTC.
A
D
B
C
AD and CB are parallel by alternate interior
angles.
A
D
B
C
If a quadrilateral has one pair of opposite sides
both congruent and parallel, it is a
parallelogram.
A
B
M
D
C
Here is a quadrilateral in which the diagonals
bisect each other.
A
B
M
D
C
The angles marked in green are congruent,
because they are vertical angles.
A
B
M
D
C
Triangle AMB is congruent to triangle CMD
by SAS.
A
B
M
D
C
The pair of sides marked in red and the pair of
angles marked in red are each congruent by
CPCTC.
A
B
M
D
C
AB is parallel to CD by alternate interior
angles.
A
B
M
D
C
The quadrilateral is parallelogram, because
one pair of opposite sides are both congruent
and parallel.
A
B
M
D
C
A quadrilateral is a parallelogram if the
diagonals bisect each other.
A
D
B
C
This quadrilateral has both pairs of opposite
angles congruent.
A
D
B
C
The triangle sum theorem tells us that the sum
of the measures of the angles of each triangle
is 180º.
A
D
B
C
Since the angles of the two triangles form the
four angles of the quadrilateral, the sum of the
measures of the four angles of the
quadrilateral must be 360º.
A
yº
xº
yº
D
B
xº
C
Let each red angle have a measure, xº, and
each green angle have a measure, yº.
A
yº
xº
yº
xº
D
C
2x + 2y = 360
B
A
yº
xº
yº
xº
D
C
2x + 2y = 360
x + y = 180
B
A
D
B
C
Each red angle is supplementary to each green
angle.
A
D
B
C
Each pair of opposite sides is parallel, because
supplementary interior angles on the same side
of a transversal give parallel lines.
The quadrilateral, therefore, is a
parallelogram.
A
D
B
C
If the opposite angles of a quadrilateral are
congruent, then the quadrilateral is a
parallelogram.
Ways to prove a quadrilateral is a parallelogram.
Ways to prove a quadrilateral is a parallelogram.
If both pairs of opposite sides of a quadrilateral are
parallel, the quadrilateral is a parallelogram.
Ways to prove a quadrilateral is a parallelogram.
If both pairs of opposite sides of a quadrilateral are
parallel, the quadrilateral is a parallelogram.
If both pairs of opposite sides are congruent, a
quadrilateral is a parallelogram.
Ways to prove a quadrilateral is a parallelogram.
If both pairs of opposite sides of a quadrilateral are
parallel, the quadrilateral is a parallelogram.
If both pairs of opposite sides are congruent, a
quadrilateral is a parallelogram.
If a quadrilateral has one pair of opposite sides
both congruent and parallel, it is a parallelogram.
Ways to prove a quadrilateral is a parallelogram.
If both pairs of opposite sides of a quadrilateral are
parallel, the quadrilateral is a parallelogram.
If both pairs of opposite sides are congruent, a
quadrilateral is a parallelogram.
If a quadrilateral has one pair of opposite sides
both congruent and parallel, it is a parallelogram.
A quadrilateral is a parallelogram if the
diagonals bisect each other.
Ways to prove a quadrilateral is a parallelogram.
If both pairs of opposite sides of a quadrilateral are
parallel, the quadrilateral is a parallelogram.
If both pairs of opposite sides are congruent, a
quadrilateral is a parallelogram.
If a quadrilateral has one pair of opposite sides
both congruent and parallel, it is a parallelogram.
A quadrilateral is a parallelogram if the
diagonals bisect each other.
If the opposite angles of a quadrilateral are
congruent, then the quadrilateral is a parallelogram.
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