The document discusses the properties of parallelograms and how to prove that a quadrilateral is a parallelogram. It states that a quadrilateral is a parallelogram if one pair of its opposite sides are parallel and congruent. It also provides shortcuts for proving that a parallelogram is a rectangle, rhombus, or square based on additional properties like right angles, congruent diagonals, or bisected angles.

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
Parallelograms
If one pair of opposite sides of a
quadrilateral is congruentcongruentcongruentcongruent and parallelparallelparallelparallel, then
the quadrilateral is a parallelogram.
We can also use the converses of the
theorems from the previous section to
prove that quadrilaterals are
parallelograms.
Parallelogram ⇔ Opposite sides ≅
Opposite angles ≅
Cons. ∠s supp.
Diagonals bisect

4. Conditions 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.