A rectangle is a parallelogram with four right angles.
While the definition contains the word "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.

The properties (theorems) will be stated in "if ...then" form. Both the theorem and its converse (where you swap the "if" and "then" expressions) will be examined.

Click in the charts below to see each proof.
The * means proof is directly referenced in Common Core.
While one method of proof will be shown, other methods are also possible.
Definition and Theorems pertaining to a rectangle:
DEFINITION: A rectangle is a parallelogram with four right angles.
 THEOREM: If a parallelogram is a rectangle, it has congruent diagonals. *
 THEOREM Converse: If a parallelogram has congruent diagonals, it is a rectangle. *

Did you know ...
the figure joining, in order, the midpoints of the sides of a rectangle is a rhombus, and the figure joining, in order, the midpoints of the sides of a rhombus is a rectangle.
 Proof 1 Proof 2

 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.

In addition, the definition could be stated as:
A rhombus is a parallelogram having two adjacent sides congruent.

Definition and Theorems pertaining to a rhombus:
DEFINITION: A rhombus is a parallelogram with four congruent sides.
 THEOREM: If a parallelogram is a rhombus, each diagonal bisects a pair of opposite angles.
 THEOREM Converse: If a parallelogram has diagonals that bisect a pair of opposite angles, it is a rhombus.
 THEOREM: If a parallelogram is a rhombus, the diagonals are perpendicular.
 THEOREM Converse: If a parallelogram has diagonals that are perpendicular, it is a rhombus.

 A square is a parallelogram with four congruent sides and four right angles.

This definition can also be stated as:
• A square is a quadrilateral that is also a rectangle and a rhombus.
• A square is a rhombus with four right angles.
• A square is a rhombus with two adjacent sides perpendicular.
• A square is a rectangle with four congruent sides.

Since we have already proven properties pertaining to the rectangle and the rhombus, no further proofs will be prepared for the square.