Rectangles, rhombuses (also called rhombi) and squares are all more specific versions of parallelograms, also called special parallelograms.
- A quadrilateral is a rectangle if and only if it has four right (congruent) angles.
- A quadrilateral is a rhombus if and only if it has four congruent sides.
- A quadrilateral is a square if and only if it has four right angles and four congruent sides. By definition, a square is a rectangle and a rhombus.
You can always show that a parallelogram is a rectangle, rhombus, or square by using the definitions of these shapes. There are some additional ways to prove parallelograms are rectangles and rhombuses, shown below:
1. A parallelogram is a rectangle if the diagonals are congruent.
2. A parallelogram is a rhombus if the diagonals are perpendicular.
3. A parallelogram is a rhombus if the diagonals bisect each angle.
What if you were given a parallelogram and information about its diagonals? How could you use that information to classify the parallelogram as a rectangle, rhombus, and/or square?
Is a rectangle SOMETIMES, ALWAYS, or NEVER a parallelogram? Explain why.
A rectangle has two sets of parallel sides, so it is ALWAYS a parallelogram.
Is a quadrilateral SOMETIMES, ALWAYS, or NEVER a pentagon? Explain why.
A quadrilateral has four sides, so it will NEVER be a pentagon with five sides.
What typed of parallelogram are the figures below?
For the second figure, all four angles are congruent but the sides are not. This is a rectangle.
Is a rhombus SOMETIMES, ALWAYS, or NEVER a square? Explain why.
A rhombus has four congruent sides and a square has four congruent sides and angles. Therefore, a rhombus is a square when it has congruent angles. This means a rhombus is SOMETIMES a square.
A square has all the properties of a parallelogram, rectangle and rhombus.
|Properties of a Parallelogram||Properties of a Rhombus||Properties of a Rectangle|
RACEis a rectangle. Find: RG AE AC EC m∠RAC
DIAMis a rhombus. Find: MA MI DA m∠DIA m∠MOA
CUBEis a square. Find: m∠UCE m∠EYB m∠UBY m∠UEB
For questions 4-15, determine if the quadrilateral is a parallelogram, rectangle, rhombus, square or none.
For questions 16-19 determine if the following are ALWAYS, SOMETIME, or NEVER true. Explain your reasoning.
- A rectangle is a rhombus.
- A square is a parallelogram.
- A parallelogram is regular.
- A square is a rectangle.
To see the Review answers, open this PDF file and look for section 6.5.