Parallelograms and Rectangles

1
Parallelograms and Rectangles
2
Quadrilateral Definitions

• Parallelogram
• opposite sides are parallel
• Rectangle
• adjacent sides are perpendicular

3

• the first proof

4
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

5

• 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

6

• 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.

7
Prove If the opposite sides are equal, then it
is a parallelogram.

• Given
• Construct segment

8

• Using the reflexive property, we can say
• Therefore, using SSS we know

9

• As the triangles are congruent, we know that
  corresponding angles are congruent.
• Therefore,
• If the alternate interior angles are congruent,
  then segments

10

• Therefore, if the opposite segments are equal,
  then it is a parallelogram.

11
Therefore

• It is a parallelogram, if and only if the
  opposite sides are equal.

12

• the second proof...

13
Prove If it is a parallelogram, then the
diagonals bisect each other.

• Given parallelogram ABCD,
• Using the property proven in the previous proof,

14
Construct segment BD

• This forms two congruent triangles,
• because of SSS, as the following segments are
  congruent
• This implies corresponding angles are congruent

15
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

16
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

17

• Since we have congruent triangles
• We can then say that
• Therefore, the diagonals of the parallelogram
  bisect each other since the segments are
  congruent.

18
ProveIf 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

19

• 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

20
therefore,

• It is a parallelogram, if and only if the
  diagonals bisect each other.

21

• the third proof

22
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.

23

• 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.

24
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)

25

• With 4 congruent triangles, we know
  corresponding sides are congruent.
• Therefore, we may state that

26

• Hence, if it is a rectangle,
• then it is a parallelogram and the diagonals are
  equal.

27
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

28

• 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.

29

• If the interior angles are 90 degrees, then we
  can say that the adjacent sides are
  perpendicular.
• Therefore, it is a rectangle.

30
THEREFORE

• It is a rectangle, if and only if it is a
  parallelogram and the diagonals are equal.

31
Parallelograms, Trapezoids, Rectangles, Rhombi,
Kites, and Squares.Oh MY!