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? After completing this Concept, you'll be able to further classify a parallelogram based on its diagonals, angles, and sides.
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.
is a rectangle if and only if .
- A quadrilateral is a rhombus if and only if it has four congruent sides.
is a rhombus if and only if .
- 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.
is a square if and only if and .
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.
is parallelogram. If , then is also a rectangle.
2) A parallelogram is a rhombus if the diagonals are perpendicular.
is a parallelogram. If , then is also a rhombus.
3) A parallelogram is a rhombus if the diagonals bisect each angle.
is a parallelogram. If bisects and and bisects and , then is also a rhombus.
What typed of parallelogram are the figures below?
a) All sides are congruent and one angle is , so the angles are not congruent. This is a rhombus.
b) 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.
List everything you know about the 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|
All the bisected angles are .
1. Is a rectangle SOMETIMES, ALWAYS, or NEVER a parallelogram? Explain why.
2. Is a rhombus SOMETIMES, ALWAYS, or NEVER equiangular? Explain why.
3. Is a quadrilateral SOMETIMES, ALWAYS, or NEVER a pentagon? Explain why.
1. A rectangle has two sets of parallel sides, so it is ALWAYS a parallelogram.
2. Any quadrilateral, including a rhombus, is only equiangular if all its angles are . This means a rhombus is SOMETIMES equiangular, only when it is a square.
3. A quadrilateral has four sides, so it will NEVER be a pentagon with five sides.
is a rectangle. Find:
is a rhombus. Find:
is a square. Find:
For questions 4-15, determine if the quadrilateral is a parallelogram, rectangle, rhombus, square or none.
For questions 16-21 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.