Some Special Parallelograms: A parallelogram is a type of quadrilateral with two pairs of parallel sides. Based on the properties of a parallelogram’s sides and/or angles, we have different special parallelograms such as rhombus, square and rectangle.

Some special parallelograms are the parallelograms with all equal sides (equilateral), all angles equal (equiangular) and both sides and angles are equal (equilateral and equiangular). Let’s learn parallelograms, their properties, and some special types.

What are Parallelograms?

A quadrilateral is a closed polygon with four sides and four angles. A parallelogram is a quadrilateral with two pairs of opposite sides that are parallel to each other. In a parallelogram, opposite angles are equal in measurement. The sum of all interior angles in the parallelogram is \(360^\circ \).

We have different types of quadrilateral such as parallelograms, squares, rhombus and rectangles. 

From the above, we can say that

1. All squares are rectangles
2. All squares are rhombus
3. All squares, rectangles, rhombus are parallelograms.
4. All parallelograms, squares, rectangles, rhombus are quadrilaterals.

Thus, quadrilaterals such as rectangles, rhombuses and squares are some special parallelograms.

Properties of Parallelograms

Parallelograms are the type of quadrilateral with two pairs of parallel sides. There are some properties associated with the parallelogram. The properties of a parallelogram are listed below:

1. In a parallelogram, the pairs of two opposite sides are parallel and equal.

In a parallelogram \(A B C D, A B || D C\) and \(A D || B C\).

2. In a parallelogram, the opposite angles are equal in measure.

In a parallelogram \(A B C D, \angle B=\angle D\) and \(\angle A=\angle C\).

3. In a parallelogram, the pair of adjacent angles are supplementary.

In a parallelogram \(A B C D, \angle A+\angle D=180^{\circ}\).

4. In a parallelogram, diagonals bisect each other.

In a parallelogram \(A B C D, B M=M D\) and \(A M=M C\).

Some Special Parallelograms

Some special parallelograms are the parallelograms with all equal sides (equilateral), all angles equal (equiangular) and both sides and angles are equal (equilateral and equiangular).

Some special parallelograms are rhombus, square and rectangle. The unique features of the special parallelograms are listed below:

1. Rhombus: A rhombus is a parallelogram with all sides of equal length.
2. Rectangle: A rectangle is a parallelogram with all angles equal in measure.
3. Square: A square is a parallelogram with all sides, and all angles are equal.

Properties of Rhombus

A two-dimensional quadrilateral with all equal sides is known as a rhombus. A parallelogram with all equal sides (equilateral) is called the rhombus. Rhombus is a special kind of parallelogram formed by making all the sides of the parallelogram with equal measurement.

Rhombus is a parallelogram in which each diagonal are perpendicular to the other. The real-life examples of the rhombus are diamond, kite etc.

Observing the rhombus \(A B C D\) as mentioned above and the unique properties of rhombus are listed below:

1. All four sides of the rhombus are equal in measure.
\(A B=B C=C D=D A\)

2. Opposite angles of the rhombus are equal in measure.
\(\angle A=\angle C\) and \(\angle B=\angle D\)

3. The diagonals \(A C\) and \(B D\) of the rhombus \(A B C D\) are perpendicular to each other.
\(A C \perp B D\)

4. Adjacent angles of the rhombus \(A B C D\) are supplementary.
\(\angle A+\angle B=\angle B+\angle C=\angle C+\angle D=\angle A+\angle D=180^{\circ}\)

5. The rhombus has two lines of symmetry.

Let “\(a\)” be the side of the rhombus and \(d_{1}, d_{2}\) are the diagonals of the rhombus, then

1. Perimeter \(=4 a\)
2. Area \(=\frac{1}{2} \times d_{1} \times d_{2}\)

Properties of Rectangle

A two-dimensional quadrilateral with opposite sides is equal and parallel, and also all angles are equal, known as a rectangle. A parallelogram with all equal angles (equiangular) is called a rectangle. A rectangle is a special kind of parallelogram formed by making any one of the angles of the parallelogram equal to the right angle \(\left({90^\circ }\right)\).

The rectangle is a parallelogram in which each diagonal are equal and bisects each other. The real-life examples of the rectangles are mobile phones, books etc.

Observing the rectangle \(A B C D\) as mentioned above and the unique properties of rectangle are listed below:

1. The opposite sides of the rectangles are parallel and also equal in measurement. \(A B || D C\) and \(A B=D C, B C || A D\) and \(B C=A D\).
2. All the angles of a rectangle are equal in measure and each being equal to the right angle.
   \(\angle A=\angle C=\angle B=\angle D=90^{\circ}\)
3. The diagonals \(A C\) and \(B D\) of the rectangle \(A B C D\) are equal in measures and bisects each other.
   \(A C=B D\) and \(A S=C S, B S=D S\).
4. Adjacent angles of the rectangle \(\mathrm{ABCD}\) are supplementary.
   \(\angle A+\angle B=\angle B+\angle C=\angle C+\angle D=\angle A+\angle D=180^{\circ}\)
5. The rectangle has two lines of symmetry.
   Let the length and breadth of the rectangle are \(l\) and \(b\), then
6. Perimeter \(=2(l+b)\)
7. Area \(=l \times b\)

Properties of Square

A two-dimensional quadrilateral with all four sides are equal, and also that all angles are equal, known as a square. A parallelogram with equal angles (equiangular) and equal sides (equilateral) is called a square. The square is a special kind of parallelogram formed by making any one of the angles of the parallelogram equal to the right angle \(\left({90^\circ }\right)\) and it is also making all of its sides equal in measurement.

A square is a parallelogram in which each diagonal are equal and bisects each other perpendicularly. The real-life examples of the squares are chess-board, the face of the cube etc.

Observing the square \(A B C D\) as mentioned above and the unique properties of square are listed below:

1. All four sides of the squares are equal in measure.
   \(A B=B C=C D=D A\)
2. All the angles of a square are equal in measure and each being equal to the right angle.
   \(\angle A=\angle C=\angle B=\angle D=90^{\circ}\)
3. The diagonals \(A C\) and \(B D\) of the square \(A B C D\) are equal in measures and perpendicular to each other.
   \(A C=B D\) and \(A C \perp B D\)
4. Adjacent angles of the square \(A B C D\) are supplementary.
   \(\angle A+\angle B=\angle B+\angle C=\angle C+\angle D=\angle A+\angle D=180^{\circ}\)
5. The square has four lines of symmetry.
   Let “\(a\)” be the side of the square, then
6. Perimeter \(=4 a\)
7. Area \(=a^{2}\)

Notes of Some Special Parallelograms

As discussed, the special parallelograms are square, rhombus and rectangle. The below table lists the properties of the special parallelograms.

Property	Rhombus	Rectangle	Square
Opposite sides are parallel and equal	Yes	Yes	Yes
All sides are equal	Yes	No	Yes
All angles are equal	No	Yes	Yes
Opposite angles are equal	Yes	Yes	Yes
Diagonals are equal	No	Yes	Yes
Diagonals are perpendicular	Yes	No	Yes
Diagonals bisect each other	Yes	Yes	Yes
Adjacent angles are supplementary	Yes	Yes	Yes

Solved Examples – Some Special Parallelograms

Q.1. In the given rectangle ABCD, the diagonals AC and BD intersect each other at point O. The value of \(BO = 15\,{\rm{cm}}\), then find the value of the diagonal AC.

Ans: In a rectangle, diagonals are equal in measure and bisect each other. \(B O=O D=15 \mathrm{~cm}\)
So, \(B D=B O+O D=15+15=30 \mathrm{~cm}\)
The rectangle is a parallelogram with the measure of diagonals are equal.
\(A C=B D=30 \mathrm{~cm}\)
Therefore, the length of the diagonal \(A C\) is \(30 \mathrm{~cm}\).

Q.2. Find the value of the “y”, the parallelogram PQRS is a rhombus.

Ans: Given that parallelogram, \(P Q R S\) is a rhombus.
We know that the rhombus is a parallelogram with all equal sides (equilateral).
\(P Q=Q R=R S=P S\)
\(\Rightarrow 2 y+3=5 y-6\)
\(\Rightarrow 5 y-2 y=6-3\)
\(\Rightarrow 3 y=3\)
\(\Rightarrow y=\frac{3}{3}=1\)
Hence, the value of the \(y=1\).

Q.3. Find the perimeter and area of the parallelogram with equal sides of \(6\,{\rm{cm}}\). All angles are right angles.
Ans: We know that the parallelogram with all four equal sides and all angles are equal to right angles is known as a square.
Given: Side of the square is \(6 \mathrm{~cm}\).
We know that the perimeter of the square is \(4 \times\) side \(=4 \times 6=24 \mathrm{~cm}\)
We know that area of the square is side \(^{2}=6^{2}=36 \mathrm{~cm}^{2}\)

Q.4. Find the value of the x in the given parallelogram.

Ans: We know that in a parallelogram, opposite angles are equal.
\(\angle B=\angle D\)
\(\Rightarrow x+29^{\circ}=87^{\circ}\)
\(\Rightarrow x=87^{\circ}-29^{\circ}\)
\(\Rightarrow x=58^{\circ}\)
Hence the value of the \(x=58^{\circ}\).

Q.5. Find the value of all the angles in the given parallelogram.

Ans: Given \(ABCD\) is a parallelogram.
The adjacent angles of a parallelogram are supplementary.
\(\angle A+\angle B=180^{\circ}\)
\(\Rightarrow x+2 x=180^{\circ}\)
\(\Rightarrow 3 x=180^{\circ}\)
\(\Rightarrow x=\frac{180^{\circ}}{3}=60^{\circ}\)
So, the value of angle \(\mathrm{A}\) is \(x=60^{\circ}\)
The value of angle \(\mathrm{B}\) is \(2 x=2 \times 60^{\circ}=120^{\circ}\)
We know that in a parallelogram, opposite angles are equal in measure.
\(\angle A=\angle C\) and \(\angle B=\angle D\)
Therefore, \(\angle C=60^{\circ}\) and \(\angle D=120^{\circ}\).
Hence, the value of all angles in a parallelogram are \(60^{\circ}, 120^{\circ}, 60^{\circ}, 120^{\circ}\).

Summary

In this article, we have discussed the definition of a parallelogram, which is a quadrilateral with pair of two equal and parallel sides. We also studied the properties of a parallelogram. This article gives some special parallelograms with equilateral, equiangular and both equilateral and equiangular.

Here, we have studied the special parallelograms such as square, rhombus and rectangle with unique properties. This article also gives the solved examples of some special parallelograms, which easily helps us understand the concept.

Learn the Properties of Parallelograms

FAQs

Q.1. What are three special parallelograms?
Ans: The three special types of parallelograms are
1. Rhombus (equilateral)
2. Rectangle (equiangular)
3. Square (equilateral and equiangular)

Q.2. How do you identify special parallelograms?
Ans: A quadrilateral, in which the opposite pair of sides are parallel, is a parallelogram. When identifying special parallelograms, we look at this basic property. We observe that a rhombus, a square and a rectangle have this property. Hence, these three quadrilaterals are called special parallelograms.

Q.3. What is the most special type of parallelogram?
Ans: Special parallelograms are square, rectangle and rhombus. We know that a square is a rhombus and a square is a rectangle. The most special type of parallelogram is a square.

Q.4. Is every rectangle a parallelogram?
Ans: Yes. Every rectangle is a parallelogram. The opposite sides of the rectangle are parallel and equal, like in a parallelogram.

Q.5. What is the difference between rhombus and square?
Ans: In a rhombus, all the angles may or may not be equal in measure, but all the angles must mandatorily be equal in a square, each being \(90^\circ \).

Now you are provided with all the necessary information on the special parallelograms and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.