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The word ‘parallelogram’ is derived from the Greek word ‘parallelogrammon’ which means “bounded by parallel lines”. A parallelogram is a quadrilateral that is bounded by parallel lines. It is a 2D shape in which the opposite sides are parallel and equal.
There are three types of parallelograms – square, rectangle, and rhombus, and each of them has its own unique properties.
Let’s understand what is a parallelogram and the properties of parallelogram.
What is a Parallelogram?
A parallelogram is a special type of quadrilateral that is formed by parallel lines. In a parallelogram, both pairs of opposite sides are parallel and equal. Hence, a parallelogram is defined as a quadrilateral in which both pairs of opposite sides are parallel and equal.
The above figure shows three types of parallelograms.
- Rectangle
- Square
- Rhombus
Properties of a Parallelogram
The following are the basic properties of parallelograms that help you identify them.
- The opposite sides of a parallelogram are parallel. Here, $\text{AB} ‖ \text{CD}$ and $\text{BC} ‖ \text{DA}$.
- The opposite sides of a parallelogram are equal. In the above figure, $\text{AB} = \text{CD}$ and $\text{BC} = \text{DA}$.
- The opposite angles of a parallelogram are equal. Here, $\angle \text{A} = \angle \text{C}$ and $\angle \text{B} = \angle \text{D}$
- The diagonals of a parallelogram bisect each other. Here, $\text{AC} = \text{BD}$
- Adjacent angles are supplementary. In the above figure, $\angle \text{A} + \angle \text{B} = 180^{\circ}$, $\angle \text{B} + \angle \text{C} = 180^{\circ}$, $\angle \text{C} + \angle \text{D} = 180^{\circ}$ and $\angle \text{D} + \angle \text{A} = 180^{\circ}$.
- The diagonals divide the parallelogram into two congruent triangles. Here, $\triangle \text{ABD} \cong \triangle \text{BCD}$, and $\triangle \text{ABC} \cong \triangle \text{ACD}$.
Diagonals of a Parallelogram Are Equal
The diagonals of a parallelogram divide it into two congruent triangles, i.e., in a parallelogram $\text{ABCD}$, $\triangle \text{ABD} \cong \triangle \text{BCD}$, and $\triangle \text{ABC} \cong \triangle \text{ACD}$.
Let’s see how to prove the above statement.
Since $\text{ABCD}$ is a parallelogram, the opposite sides are equal.
Therefore, $\text{AB} = \text{CD}$ and $\text{BC} = \text{DA}$.
Now, in $\triangle \text{ABD}$ and $\triangle \text{CBD}$
$\text{AB} = \text{CD}$ (Opposite sides of a parallelogram)
$\text{DA} = \text{BC}$ (Opposite sides of a parallelogram)
$\text{BD} = \text{BD}$ ( common)
Thus, $\triangle \text{ABD} \cong \triangle \text{CBD}$ (SSS congruency criterion).
Therefore, we can say that the diagonal of a parallelogram divides it into two congruent triangles.
Opposite Sides of a Parallelogram Are Equal
The opposite sides of a parallelogram are equal, i.e., in a parallelogram $text{ABCD}$, $\text{AB} = \text{CD}$, and $\text{BC} = \text{DA}$.
Let’s see how to prove the above statement.
In the above figure, $\text{ABCD}$ is a parallelogram, and $\text{AC}$ is one of the diagonals.
The diagonal $\text{AC}$ divides parallelogram $\text{ABCD}$ into two triangles, namely, $\triangle \text{ABC}$ and $\triangle \text{ABC}$.
In order to prove that opposite sides are equal i.e., $\text{AB} = \text{CD}$, and $\text{BC} = \text{DA}$, we need to first prove that $\triangle \text{ABC} \cong \triangle \text{ABC}$
In $\triangle \text{ABC}$ and $\triangle \text{CDA}$, $\text{BC} || \text{DA}$ and $\text{AC}$ is a transversal.
So, $\angle \text{BCA} = \angle \text{DAC}$ (Pair of alternate angles)
And $\text{AC} = \text{CA}$ (common)
Thus, $\triangle \text{ABC} \cong \triangle \text{CDA}$ (ASA rule).
Therefore, the corresponding parts $\text{AB} = \text{CD}$ and $\text{DA}= \text{BC}$.
Opposite Angles of a Parallelogram Are Equal
In a parallelogram the opposite angles are equal.
Let’s see how to prove the above statement, i.e, in a parallelogram $\text{ABCD}$, $\angle \text{A} = \angle \text{C}$, and $\angle \text{B} = \angle \text{D}$.
In the above figure, in a parallelogram $\text{ABCD}$, $\text{AB} || \text{CD}$ and $\text{AD} || \text{BC}$.
Consider $\triangle \text{ABC}$ and $\triangle \text{ADC}$
$\text{AC} = \text{AC}$ (common side)
We know that alternate interior angles are equal.
$\angle \text{BAC} = \angle \text{ACD}$
$\angle \text{BCA} = \angle \text{CAD}$
Therefore, $\triangle \text{ABC} \cong \triangle \text{ADC}$
Hence, $\angle \text{A} = \angle \text{C}$, and $\angle \text{B} = \angle \text{D}$. (Corresponding Parts of Congruent Triangles).
Adjacent Angles of a Parallelogram Are Supplementary
In a parallelogram the adjacent angles are supplementary.
Let’s see how to prove the above statement, i.e, in a parallelogram $\text{ABCD}$, $\angle \text{A} + \angle \text{B} = 180^{\circ}$, and $\angle \text{C} + \angle \text{D} = 180^{\circ}$.
In the above figure, $\text{AB} ∥ \text{CD}$ and $\text{AD}$ is a transversal.
We know that interior angles on the same side of a transversal are supplementary.
Therefore, $\angle \text{A} + \angle \text{D} = 180^{\circ}$
Similarly, $\angle \text{B} + \angle \text{C} = 180^{\circ}$, $\angle \text{C} + \angle \text{D} = 180^{\circ}$ and $\angle \text{A} + \angle \text{B} = 180^{\circ}$.
Therefore, the sum of any two adjacent angles of a parallelogram is equal to $180^{\circ}$.
Practice Problems
- Define parallelogram.
- State True or False
- Adjacent sides of a parallelogram are equal
- Adjacent sides of a parallelogram are parallel
- Opposite sides of a parallelogram are equal
- Opposite sides of a parallelogram are parallel
- Adjacent angles of a parallelogram are equal
- Adjacent angles of a parallelogram are supplementary
- Opposite angles of a parallelogram are equal
- Opposite angles of a parallelogram are supplementary
FAQs
What is a parallelogram in geometry?
In geometry, a parallelogram is a quadrilateral (4-sided 2D shape) that has its opposite sides parallel and equal in length.
Are all the angles of a parallelogram equal?
No, all the angles of a parallelogram are not equal. Only the opposite angles of a parallelogram are equal, whereas the adjacent angles of a parallelogram are supplementary, i.e., their sum is $180^{\circ}$.
What is the difference between a parallelogram and a quadrilateral?
All parallelograms are quadrilaterals but all quadrilaterals are not necessarily parallelograms. For example, a trapezium is a quadrilateral, but not a parallelogram. For a quadrilateral to be a parallelogram, all the opposite sides must be parallel and equal to each other.
Is a rhombus a parallelogram?
Yes, a rhombus is a parallelogram in which the opposite sides are parallel and the opposite angles are congruent. Apart from this, all the sides of a rhombus are equal and the diagonals bisect each other at right angles.
Is a trapezium a parallelogram?
No, a trapezium is not a parallelogram, since all opposite sides of the trapezium are not parallel to each other. A trapezium has only one pair of opposite sides parallel to each other. Also, a trapezium doesn’t have opposite sides equal to each other. Hence, it is a quadrilateral but not a parallelogram.
Conclusion
A parallelogram is a special type of quadrilateral that is formed by parallel lines. In a parallelogram, both pairs of opposite sides are parallel and equal. A parallelogram has certain unique properties that distinguish it from the other quadrilaterals. The three types of parallelograms are square, rectangle, and rhombus.
Recommended Reading
- How to Construct a Triangle(With Steps, Diagrams & Examples)
- Median of a Triangle(Definition & Properties)
- Altitude of a Triangle(Definition & Properties)
- Congruence of Triangles Criteria – SSS, SAS, ASA, RHS
- Similarity of Triangles Criteria – SSS, SAS, AA
- Angle Bisector of a Triangle – Definition, Properties & Examples
- What is Quadrilateral in Math(Definition, Shape & Examples)
- Properties of Triangle – Theorems & Examples
- How to Construct a Triangle(With Steps, Diagrams & Examples)
- Median of a Triangle(Definition & Properties)
- Altitude of a Triangle(Definition & Properties)
- Congruence of Triangles Criteria – SSS, SAS, ASA, RHS
- Similarity of Triangles Criteria – SSS, SAS, AA
- Types of Triangles – Definition & Examples
- What is Triangle in Geometry – Definition, Shapes & Examples
- Pair of Angles – Definition, Diagrams, Types, and Examples
- Construction of Angles(Using Protractor & Compass)
- Types of Angles in Maths(Acute, Right, Obtuse, Straight & Reflex)
- What is an Angle in Geometry – Definition, Properties & Measurement
- How to Construct a Tangent to a Circle(With Steps & Pictures)
- Tangent of a Circle – Meaning, Properties, Examples
- Angles in a Circle – Meaning, Properties & Examples
- Chord of a Circle – Definition, Properties & Examples
- How to Draw a Circle(With Steps & Pictures)
- What is a Circle – Parts, Properties & Examples
- How to Construct a Perpendicular Line (With Steps & Examples)
- How to Construct Parallel Lines(With Steps & Examples)
- How To Construct a Line Segment(With Steps & Examples)
- What are Collinear Points in Geometry – Definition, Properties & Examples
- What is a Transversal Line in Geometry – Definition, Properties & Examples
- What are Parallel Lines in Geometry – Definition, Properties & Examples
- What is Concurrent lines in Geometry – Definition, Conditions & Examples
- What is Half Line in Geometry – Definition, Properties & Examples
- What is a Perpendicular Line in Geometry – Definition, Properties & Examples
- Lines in Geometry(Definition, Types & Examples)
