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. Since these are congruent triangles, we may assume that
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. If the alternate interior angles are congruent, then segments
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. It is a parallelogram, if and only if the opposite sides are equal.
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. 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

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. It is a parallelogram, if and only if the diagonals bisect each other.
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…
Parallelograms, Trapezoids, Rectangles, Rhombi, Kites, and Squares….Oh MY!