Parallelograms and Rectangles
Quadrilateral Definitions
Parallelogram: opposite sides are parallel
Rectangle: adjacent sides are
perpendicular
the first proof…
Prove: If it is a parallelogram, then the opposite sides are
equal.By definition, a parallelogram has opposite sides that are parallel.
Construct a segment:
D
B C
A
AC
We may use the properties of parallel lines to show certain angle congruencies.
As they are alternate interior angles,
and using the reflexive property, we know
D
B C
A
DCABACandDACBCA
CACA
Therefore, we know that the following triangles are congruent because of
ASA
Since these are congruent triangles, we may assume that
Therefore, if it is a parallelogram, then the opposite sides are equal.
D
B C
A
DCABAC
DCBAandDABC
Prove: If the opposite sides are equal, then it is a
parallelogram.
Given:
Construct segment
D
B C
A
DCBAandDABC
AC
Using the reflexive property, we can say
Therefore, using SSS we know
D
B C
A
ACAC
CDAABC
As the triangles are congruent, we know that corresponding angles are congruent.
Therefore,
If the alternate interior angles are congruent, then segments
D
B C
A
DACBCAandDCABAC
DCBAandDABC
Therefore, if the opposite segments are equal, then it is
a parallelogram.
D
B C
A
Therefore…
It is a parallelogram, if and only if the opposite sides are equal.
the second proof...
Prove: If it is a parallelogram, then the
diagonals bisect each other.Given parallelogram ABCD,
Using the property proven in the previous proof,
D
B C
A
BCADandDCAB
Construct segment BD
This forms two congruent triangles,
because of SSS, as the following segments are congruent:
This implies corresponding angles are congruent
D
B C
A
CDBABD
BDBDADBCCDAB ,,
Construct segment AC
This also forms two congruent triangles
Because of SSS, as the following sides are congruent
This implies that corresponding angles are congruent
D
B C
A
CDAABC
ACACADBCCDAB ,,
Look at both diagonals and the created triangles
With both diagonals displayed, we may conclude that we have two sets of congruent triangles, based upon ASA.For example,
since
E
D
B C
A
DAEBCE
EADECBDABCEDAEBC ,,
Since we have congruent triangles
We can then say that
Therefore, the diagonals of the parallelogram bisect each other since
the segments are congruent.
DAEBCE
E
D
B C
A
AECEandDEBE
Prove:If the diagonals bisect each other, then it is a
parallelogram.
Since the diagonals bisect each other, we know certain segments are congruent.
We may also say that vertical angles are congruent
E
D
B C
A
AECEandDEBE
DECBEAandCEBAED
Using SAS, we may say there are two sets of congruent triangles
Therefore, we may say
Therefore, since the diagonals bisect each other, then the opposite sides are congruent. From the previous proof, we know that it is a parallelogram
E
D
B C
A
DECBEAandDAEBCE
BCADandDCAB
therefore,
It is a parallelogram, if and only if the diagonals bisect each other.
the third proof…
Prove: If it is a rectangle, then it is a parallelogram and the diagonals are
equal.
By definition, a rectangle has adjacent sides that are perpendicular.
Since segment BC and segment AD are both perpendicular to segment AB, we may conclude that segment BC and segment AD are parallel.
The same may be concluded about segments AB and DC.
DA
B C
Since opposite sides are parallel, we may conclude that the rectangle is also a parallelogram.
Since it is a parallelogram, then we know that opposite sides are congruent.
DA
B C
Construct Segments AC and BD
Since the rectangle is also a parallelogram, then we may say,
With the constructed segments, the congruent sides, and the right angles, we have 4 congruent triangles (by SAS):
E
DA
B C
CDABandBDAC
CDBCDAABDABC
With 4 congruent triangles, we know corresponding sides are congruent.
Therefore, we may state that:
E
DA
B C
BDAC
Hence, if it is a rectangle,
then it is a parallelogram and the diagonals are equal.
Prove: If it is a parallelogram and the diagonals are equal,
then it is a rectangle.
Given: Opposite sides of a parallelogram are both parallel and congruent.Given: The diagonals are equal.
Using SSS, we know the 4 following triangles are congruent:
E
DA
B C
CDBCDAABDABC
If the four triangles are congruent, then corresponding angles are congruent.
The sum of the angles in the parallelogram (or any quadrilateral for that matter) must be 360 degrees, and all of the interior angles must be congruent.
904
360
If the interior angles are 90 degrees, then we can say that the adjacent sides are perpendicular.
Therefore, it is a rectangle.
THEREFORE…
It is a rectangle, if and only if it is a parallelogram and the diagonals are
equal.
Parallelograms, Trapezoids, Rectangles, Rhombi, Kites, and
Squares….Oh MY!
Rectangle
Paralleogram
TrapezoidSquare Kite
Quadrilateral
Rhombus
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