Theorems Dealing with Rectangles, Rhombuses, Squares
Rectangle
Definition: A rectangle is a parallelogram with four right angles.
Properties:
- Rectangle has all of the properties of the parallelogram.
- 4 right angles
- diagonals congruent
Using the definition, the properties of the rectangle can be “proven” true and become theorems.
When dealing with a rectangle, the definition and theorems are stated as …
- A rectangle is a parallelogram with four right angles.
While the definition states “parallelogram”, it is sufficient to say: “A quadrilateral is a rectangle if and only if it has four right angles.”, since any quadrilateral with four right angles is a parallelogram. - If a parallelogram has one right angle it is a rectangle.
- A parallelogram is a rectangle if and only if its diagonals are congruent.
Construction workers use the fact that the diagonals of a rectangle are congruent (equal) when attempting to build a “square” footing for a building, a patio, a fenced area, a table top, etc. Workers measure the diagonals. When the diagonals of the project are equal the building line is said to be square.
Rhombus
Definition: A rhombus is a parallelogram with four congruent sides.
Properties:
- Rhombus has all of the properties of the parallelogram.
- 4 congruent sides
- diagonals bisect angles
- diagonals perpendicular
Using the definition, the properties of the rhombus can be “proven” true and become theorems.
When dealing with a rhombus, the definition and theorems are stated as …
- A rhombus is a parallelogram with four congruent sides.
While the definition states “parallelogram”, it is sufficient to say: “A quadrilateral is a rhombus if and only if it has four congruent sides.”, since any quadrilateral with four congruent sides is a parallelogram. - If a parallelogram has two consecutive sides congruent, it is a rhombus.
- A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.
- A parallelogram is a rhombus if and only if the diagonals are perpendicular. (Proof of theorem appears further down page.)
Square
Definition: A square is a parallelogram with four congruent sides and four right angles.
Square has all of the properties of the parallelogram AND the rectangle AND the rhombus.
Using the definition, the properties of the rhombus can be “proven” true and become theorems.
When dealing with a square, the definition is stated as …
A square is a parallelogram with four congruent sides and four right angles.
This definition may also be stated as A quadrilateral is a square if and only if it is a rhombus and a rectangle.
Proof of Theorem: If a parallelogram is a rhombus, then the diagonals are perpendicular.
(Remember: when attempting to prove a theorem to be true, you cannot use the theorem as a reason in your proof.)