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Special Parallelograms: Rhombus, Square & Rectangle
A parallelogram is a quadrilateral in which the opposite sides are parallel and equal, and the opposite angles are of equal measure. Parallelograms can be equilateral (with all sides of equal length), equiangular (with all angles of equal measure), or, both equilateral and equiangular. Let us learn more about the three special parallelograms: rhombus, square, and rectangle along with their properties.
|What are Parallelograms?
|Properties of Special Parallelograms
|FAQs on Special Parallelograms: Rhombus, Square & Rectangle
What are Parallelograms?
A parallelogram is a two-dimensional quadrilateral with two pairs of parallel sides. The opposite angles and opposite sides of a parallelogram are congruent and the sum of its interior angles is 360°.
Relationship Between Various Quadrilaterals and Parallelograms
Quadrilaterals like rhombi (plural for rhombus), squares, and rectangles have all the properties of a parallelogram. Observe the following figure which shows the relationship between various quadrilaterals and parallelograms.
If we observe the figure shown above, we understand that:
- Every square is a rectangle.
- Every square is a rhombus.
- Every rhombus, square and rectangle is a parallelogram.
- All parallelograms are quadrilaterals.
Now, let us learn about some special parallelograms.
A rhombus, a rectangle, and a square are special parallelograms because they not only show the properties of a parallelogram but also have unique properties of their own. Let us have a look at the unique features of special parallelograms.
- A rhombus, which is also called a diamond, is a special parallelogram with four congruent sides.
- A rectangle is a special parallelogram in which all four angles are equal to 90°.
- A square is a special parallelogram that is both equilateral and equiangular.
Properties of Special Parallelograms
Each special parallelogram has specific properties of its own. Let us learn about each of them in this section.
A rhombus is a two-dimensional quadrilateral in which all the sides are equal and the opposite sides are parallel. It is a parallelogram whose diagonals are perpendicular to each other. Some of the real-life examples of a rhombus are kite, diamond, etc.
Observe the rhombus ABCD and note the properties listed below:
- All sides are congruent. Side AB = BC = CD = DA
- Opposite angles are congruent. Angles ∠A = ∠C and ∠B = ∠D
- The diagonals AC and BD bisect each other at right angles.
- Adjacent angles in a rhombus are supplementary (For example, ∠A + ∠B = 180°).
A square is a two-dimensional quadrilateral with four equal sides and four equal angles. It is a special parallelogram in which all angles and sides are equal. Some of the real-life examples of a square are a slice of bread, chessboard etc.
Observe the square GDEF and note the properties listed below:
- All sides are congruent. Sides GF = FE = ED = DG
- All the angles are 90°. Angles ∠G = ∠F = ∠E = ∠D = 90°.
- The opposite sides are parallel to each other. GF || DE and GD || FE
- The diagonals are congruent. Diagonals GE = FD
- The diagonals GE and FD bisect each other at right angles.
A rectangle is a two-dimensional quadrilateral in which the opposite sides are equal and parallel and all its angles are equal. Some of the real-life examples of a rectangle are books, mobile phones, etc.
Observe the rectangle MNOP and note the properties listed below:
- The opposite sides are parallel. MN || PO and MP || NO
- The opposite sides are congruent. MN = PO and MP = NO
- All angles are right angles. ∠M = ∠N = ∠O = ∠P = 90°
- The diagonals MO and PN are congruent and bisect each other.
Summary of the Properties
The following table shows a summary and a comparison of the properties of special parallelograms: rhombus, square & rectangle.
Check out these interesting articles to learn more about the properties of special parallelograms and their related topics
- Types of Triangles
- Properties of Triangle
- What are the Properties of a Parallelogram?
- Properties of Rectangle
Here is a list of a few points that should be remembered while studying about parallelograms:
- A quadrilateral is a four-sided two-dimensional figure whose interior angles sum up to 360°.
- A parallelogram can be defined as a quadrilateral with four sides in which two sides are parallel to each other.
- A square is a special parallelogram that is both equilateral and equiangular and with diagonals perpendicular to each other.
- A rectangle is a special parallelogram whose opposite sides are congruent and each angle is equal to 90°.
- A rhombus, which is also called a diamond, is a special parallelogram with four congruent sides with diagonals perpendicular to each other.
Example 1: In the given rectangle EFGH, diagonals EG and FH intersect at point O. If EO = 16 units, then find FH.
As per the properties of a rectangle, the diagonals of a rectangle bisect each other. Hence, we can say that EO = GO.
Since the diagonals are congruent, EG = FH
EO = 16, and GO = 16
EG = EO + OG
EG = 16 + 16
EG = 32
Since EG = FH
Therefore, FH = 32 units.
Example 2: For square PQRS, state whether the following statements are true or false.
- The length of PR is equal to the length of SQ.
- The diagonals PR and SQ bisect each other at right angles.
- The perimeter of the above square could be given as 4SR.
1. The length of PR equal the length of SQ – True
Reason: Diagonals of a square are congruent.
2. The diagonals PR and SQ bisect each other at right angles – True
Reason: Diagonals of a square always bisect each other at right angles.
3. The perimeter of the above square could be given as 4SR – True
Reason: All sides of a square are congruent. Perimeter is defined as the sum of all the sides of a closed figure. For square PQRS, perimeter = PQ + QR + RS + SP. Since all the four sides in a square are congruent, PQ = QR = RS = SP, the perimeter could be given as four times of any one side of the square, say SR. Thus, the perimeter of the above square could be given as 4SR.
FAQs on Special Parallelograms: Rhombus, Square & Rectangle
What Is the Difference Between a Parallelogram, a Square, and a Rhombus?
The following points show the basic difference between a parallelogram, a square, and a rhombus:
- In a parallelogram, the opposite sides are parallel and equal.
- In a rhombus, all four sides are of the same length and its opposite sides are parallel.
- In a square, all four sides are of the same length and all angles are equal to 90°.
When Can a Rhombus Become a Rectangle?
A rhombus can become a rectangle only if all four angles of the rhombus are 90°.
What Is the Sum of the Interior Angles of a Quadrilateral?
The sum of the interior angles of a quadrilateral is equal to 360°.
Which Parallelogram Is Both a Rectangle and a Rhombus?
If a parallelogram is both a rectangle and a rhombus, then all its sides should be equal and all its angles should be equal to 90°. This holds true for a square.Therefore, a square can be a rectangle and a rhombus.
Name 3 Special Parallelograms.
The 3 special parallelograms are rectangle, square, and rhombus.
What Are the Different Types of Quadrilaterals?
The different types of quadrilaterals are– parallelogram, trapezium or trapezoid, rectangle, square, kite, and rhombus.
Is Every Rectangle a Parallelogram?
Yes, every rectangle is a parallelogram since the opposite sides of rectangles are parallel and equal.