4.3 Rhombuses, Rectangles and Squares
Under the category of parallelograms, there are some parallelograms with special characteristics.
- Definition: A rectangle is a parallelogram with four right angles.
- Definition: A rhombus is a parallelogram with all four sides congruent.
- Definition: A square is a parallelogram with four right angles and all four sides congruent.
Notice that a rectangle, rhombus and a square are all parallelograms, so that everything that holds true for a parallelogram will also hold true for each of these quadrilaterals. There are other special properties that hold true for each of these three quadrilaterals and we shall develop and prove theorems to support these properties.
- Theorem 4.10: If a quadrilateral is a rectangle, then the diagonals are congruent.
- Theorem 4.11: If a quadrilateral is a rhombus, then the diagonals are perpendicular to each other.
- Theorem 4.12:
If a quadrilateral is a rhombus, then the diagonals bisect the angles
of the rhombus.
The proofs of these three theorems will be left as exercises.
It is important to note that since a square has four right angles it is a rectangle, and because all four sides of a square are congruent it is also a rhombus. Therefore all of the properties of a rectangle and a rhombus also hold true for a square.