Transcript Parallelograms and Rectangles

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.
C
B
A
D
Construct a segment:
AC
We may use the properties of parallel lines to show certain angle
congruencies.
C
B
A
D
As they are alternate interior angles,
 BCA   DAC and  BAC   DCA
and using the reflexive property, we know
CA  CA
Therefore, we know that the following triangles are congruent
because of ASA
C
B
 BAC   DCA
A
D
Since these are congruent triangles, we may assume that
BC  DA and BA  DC
Therefore, if it is a parallelogram, then the opposite sides are equal.
Prove: If the opposite sides are equal,
then it is a parallelogram.
Given:
BC  DA and BA  DC
C
B
A
Construct segment
D
AC
Using the reflexive property, we can say
AC  AC
C
B
A
D
Therefore, using SSS we know
 ABC   CDA
As the triangles are congruent, we know that corresponding angles
are congruent.
C
B
A
D
Therefore,
 BAC   DCA and  BCA   DAC
If the alternate interior angles are congruent, then segments
BC DA and BA DC
Therefore, if the opposite segments are
equal, then it is a parallelogram.
C
B
A
D
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,
C
B
A
D
Using the property proven in the previous proof,
AB  DC
and
AD  BC
Construct segment BD
C
B
A
This forms two congruent triangles,
D
 ABD   CDB
because of SSS, as the following segments are congruent:
AB  CD , BC  AD , BD  BD
This implies corresponding angles are congruent
Construct segment AC
C
B
A
D
This also forms two congruent triangles
 ABC   CDA
Because of SSS, as the following sides are congruent
AB  CD , BC  AD , AC  AC
This implies that corresponding angles are congruent
Look at both diagonals and the created
triangles
C
B
E
A
D
With both diagonals displayed, we may conclude that we have two sets of
congruent triangles, based upon ASA.
For example,
since
 BCE   DAE
 EBC   EDA , BC  DA ,  ECB   EAD
Since we have congruent triangles
 BCE   DAE
C
B
E
A
D
We can then say that
BE  DE
and
CE  AE
Therefore, the diagonals of the parallelogram bisect each other since
the segments are congruent.
Prove:If the diagonals bisect each
other, then it is a parallelogram.
C
B
E
A
D
Since the diagonals bisect each other, we know certain segments are
congruent.
BE  DE
and
CE  AE
We may also say that vertical angles are congruent
 AED   CEB and  BEA   DEC
C
B
E
A
D
Using SAS, we may say there are two sets of congruent triangles
 BCE   DAE
Therefore, we may say
and
 BEA   DEC
BCthe opposite sides
Therefore,AB
since 
theDC
diagonalsand
bisect eachAD
other,then
are congruent. From the previous proof, we know that it is a
parallelogram
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.
B
C
A
D
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.
Since opposite sides are parallel, we may conclude that the
rectangle is also a parallelogram.
B
C
A
D
Since it is a parallelogram, then we know that opposite sides are
congruent.
Construct Segments AC and BD
C
B
E
A
D
Since the rectangle is also a parallelogram, then we may say,
AC  BD and
AB  CD
With the constructed segments, the congruent
sides, and the
right angles, we have 4 congruent
triangles (by SAS):
 ABC   ABD   CDA   CDB
C
B
E
A
D
With 4 congruent triangles, we know corresponding sides are
congruent.
Therefore, we may state that:
AC  BD
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.
C
B
E
A
D
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:
 ABC   ABD   CDA   CDB
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.
360
4

 90

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!
Quadrilateral
Paralleogram
Rhombus
Trapezoid
Square
Rectangle
Kite