Key Points

A quadrilateral is a polygon with four sides, four angles, and four vertices. A parallelogram is a special quadrilateral where both pairs of opposite sides are parallel.

Theorem 8.1 states that a diagonal of a parallelogram divides it into two congruent triangles. For parallelogram ABCD, diagonal AC creates $\triangle ABC \cong \triangle CDA$.

Theorem 8.2 states that in a parallelogram, opposite sides are equal. If ABCD is a parallelogram, then $AB = DC$ and $AD = BC$.

Theorem 8.4 states that in a parallelogram, opposite angles are equal. If ABCD is a parallelogram, then $\angle A = \angle C$ and $\angle B = \angle D$.

Theorem 8.6 states that the diagonals of a parallelogram bisect each other. If diagonals AC and BD intersect at O, then $OA = OC$ and $OB = OD$.

Theorem 8.3 is the converse of Theorem 8.2. If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.

Theorem 8.5 is the converse of Theorem 8.4. If each pair of opposite angles in a quadrilateral is equal, then it is a parallelogram.

Theorem 8.7 is the converse of Theorem 8.6. If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

A rectangle is a parallelogram with one right angle ($90^{\circ}$), which implies all four angles are right angles. Its diagonals are equal and bisect each other.

A rhombus is a parallelogram with all four sides of equal length. Its diagonals are perpendicular bisectors of each other.

A square is a rectangle with equal adjacent sides, meaning it has all the properties of a parallelogram, rectangle, and rhombus. Its diagonals are equal and bisect each other at right angles.

Theorem 8.8 states that the line segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half its length. If E and F are mid-points of sides AB and AC, then $EF \| BC$ and $EF = \frac{1}{2} BC$.

Theorem 8.9 states that a line drawn through the mid-point of one side of a triangle, parallel to another side, bisects the third side.