Rectangles and Squares

The Rectangle

A Rectangle is a quadrilateral in which:

1. The angles are all right angles ($90^\circ$).
2. The opposite sides are of equal length and parallel.

Diagonal Properties

A defining characteristic of a rectangle is found in its diagonals. By using the SAS (Side-Angle-Side) congruence condition on the triangles formed by the diagonals, we can deduce:

1. Lengths are Equal: The diagonals of a rectangle are of equal length ($AC = BD$).
2. Bisect Each Other: The diagonals intersect at their midpoints ($OA = OC$ and $OB = OD$).

The Square

A Square is a special type of rectangle.

• Definition: A quadrilateral in which all angles are $90^\circ$ and all sides are of equal length.
• Venn Relationship: Every square is a rectangle, but not every rectangle is a square.

Diagonal Properties of a Square

Since a square is a rectangle, its diagonals are equal and bisect each other. However, it has extra properties:

1. Perpendicular Intersection: The diagonals intersect at $90^\circ$.
2. Angle Bisectors: The diagonals bisect the angles of the square (each part is $45^\circ$).