Prove that a given quadrilateral is a parallelogram or special parallelogram

1. Conditions for Parallelograms
The student is able to (I can):
• Prove that a given quadrilateral is a parallelogram.
• Prove that a given quadrilateral is a rectangle, rhombus, or
square.

2. Recall that a parallelogram has the following properties:
We can use these properties to prove that a given
quadrilateral is a parallelogram.
Opposite sides are parallel. (Definition)
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.

3. Conditions for ParallelogramsConditions for ParallelogramsConditions for ParallelogramsConditions for Parallelograms
We can use the converses of the theorems from the previous
section to prove that quadrilaterals are parallelograms.
Parallelogram ⇔ Opposite sides (definition)
Opposite sides ≅
Opposite angles ≅
Cons. ∠s supp.
Diagonals bisect each other
In addition:
If one pair of opposite sides of a quadrilateral is congruentcongruentcongruentcongruent
and parallelparallelparallelparallel, then the quadrilateral is a parallelogram.

4. Conditions for Special ParallelogramsConditions for Special ParallelogramsConditions for Special ParallelogramsConditions for Special Parallelograms
You can always use the definitions to prove that a
quadrilateral is a special parallelogram, but there are also
some shortcuts we can use.
For all of these shortcuts, we must first prove or know that
the quadrilateral is a parallelogram.
• To prove a parallelogramparallelogramparallelogramparallelogram is a rectangle (pick one):
– One angle is a right angle
– The diagonals are congruent

5. • To prove a parallelogramparallelogramparallelogramparallelogram is a rhombus (pick one):
– A pair of consecutive sides is congruent
– The diagonals are perpendicular
– One diagonal bisects a pair of opposite angles
• To prove that a quadrilateral is a square:
– It is both a rectangle and a rhombus.