### Theorems about
Parallelograms

**Definition 1**: A
**parallelogram**
is a four sided figure where the opposite sides are parallel.

**Theorem 1**: In a parallelogram, the
opposite sides are of equal length.

**Theorem 2**: If the opposite sides in a
quadrilateral are the same length, then the figure is a
parallelogram.

**Theorem 3**: A quadrilateral is a
parallelogram if and only if the diagonals bisect each other.

**Theorem 4**: If one pair of opposite
sides in a four sided figure are both opposite and parallel, then the
figure is a parallelogram.

**Definition 2**: A
**rectangle** is a
quadrilateral where all four angles are the same size.

**Theorem 5**: A rectangle is a
parallelogram.

**Theorem 6**: A rectangle is a
parallelogram if and only if the diagonals are the same length.

**Definition 3**: A
**rhombus**
is a quadrilateral where all four sides have the same length.

**Theorem 7**: A rhombus is a
parallelogram.

**Theorem 8**: A quadrilateral is a
rhombus if and only if the diagonals are perpendicular bisectors of
each other.

**Theorem 9**: A quadrilateral is a
rhombus if and only if the diagonals bisect all the vertex angles.

Theorem 10 : If one of the diagonals in a parallelogram bisects one of the vertex angles, then the parallelogram is a rhombus.