Properties of Rectangle – Angles, Sides & Diagonals

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A rectangle is a closed 2D shape with four sides, four angles, and four vertices. It’s a type of quadrilateral and parallelogram whose measure of all angles is $90^{\circ}$. The word ‘rectangle’ starts with the Latin word ‘rectus’, which means ‘right’ or ‘straight’.

Let’s understand what are the properties of rectangle with examples.

Properties of a Rectangle

The following are the most important properties of a rectangle.

• A rectangle is a quadrilateral.
• The opposite sides of a rectangle are equal and parallel to each other.
• The interior angle of a rectangle at each vertex is $90^{\circ}$.
• The sum of all interior angles is $360^{\circ}$.
• The diagonals bisect each other.
• The length of the diagonals is equal.
• The length of the diagonal with sides a and b is, diagonal = $\sqrt{a^{2} + b^{2}}$
• Properties of a Rectangle
• The following are the most important properties of a rectangle.
• A rectangle is a quadrilateral.
• The opposite sides of a rectangle are equal and parallel to each other.
• The interior angle of a rectangle at each vertex is $90^{\circ}$.
• The sum of all interior angles is $360^{\circ}$.
• The diagonals bisect each other.
• The length of the diagonals is equal.
• The length of the diagonal with sides a and b is, diagonal = $\sqrt{a^{2} + b^{2}}$

Diagonal of a Rectangle

The line segments that join the opposite corners of a rectangle are called diagonals.

In the above figure, the two diagonals of the rectangle $\text{ABCD}$ are $\text{AC}$ and $\text{BD}$. The diagonals of a rectangle are of the same length., i.e., in the rectangle $\text{ABCD}$, $\text{AC} = \text{BD}$.

If we know the length and width of a rectangle, we can calculate the diagonal of a rectangle using the Pythagorean Theorem. 

In the above figure, $\angle \text{ADB}$ is right-angled at $\text{A}$. The diagonal $\left( \text{BD} \right)$ of the rectangle forms its hypotenuse. 

So, using the Pythagorean theorem, we get, $\text{Diagonal}^2 = \text{Length}^2 + \text{Width}^2$

Therefore, $\text{Diagonal } = \sqrt{\text{Length}^2 + \text{Width}^2}$. 

The Length of the Diagonals of a Rectangle Is Equal

Consider a rectangle $\text{ABCD}$, with diagonals $\text{AC}$, and $\text{BD}$.

Now, consider $\triangle \text{ABC}$ and $\triangle \text{DCB}$.

$\angle \text{ABC} = \angle \text{DCB} = 90^{\circ}$ (Angles of rectangle)

$\text{BC} = \text{BC}$  (common side)

$\text{AB} = \text{DC}$  (Opposite sides of a parallelogram are equal)

Therefore, $\triangle \text{ABC} \cong \triangle \text{DCB}$ (SAS Congruence Criterion)

Thus, $\text{AC} = \text{BD}$  (Corresponding Parts of Congruent Triangles)

Hence the diagonals of a rectangle are equal.

Diagonals of a Rectangle Bisect Each Other

Consider a rectangle $\text{ABCD}$ with diagonals $\text{AC}$ and $\text{BD}$ intersecting at $\text{O}$.

In $\triangle \text{OAD}$ and $\triangle \text{OCB}$,

$\angle \text{ODA} = \angle \text{OBC}$ (Alternate interior angles as $\text{AD} || \text{BC}$ and $\text{BD}$ is a transversal)

$\text{AD} = \text{BC}$ (Opposite sides of a rectangle are equal)

$\angle \text{OAD} = \angle \text{OCB}$ (Alternate interior angles as $\text{AD} || \text{BC}$ and $\text{AC}$ as transversal)

Therefore, $\triangle \text{OAD} \cong \triangle \text{OCB}$  (ASA Congruence Criterion)

Equating the corresponding parts of congruent triangles, we get:

Hence, $\text{AO} = \text{CO}$ and $\text{BO} = \text{DO}$ (Corresponding Parts of Congruent Triangles)

Thus, the diagonals of a rectangle bisect each other.

What is a Rectangle?

A rectangle is a closed 2D shape, having four sides, four angles, and four vertices. Each of the four angles of a rectangle is a right angle, i.e., measure $90^{\circ}$. The opposite sides of a rectangle are equal and parallel. Since a rectangle is a 2D shape, it is characterized by two dimensions, length, and width. Length is the longer side of the rectangle and width is the shorter side.

In the above figure, $\text{ABCD}$ is a rectangle. The longer sides $\text{AB}$ and $\text{CD}$ are called length and the shorter sides $\text{BC}$ and $\text{AD}$ are called width(or breadth). The four angles of the rectangle $\text{ABCD}$ are $\angle \text{A}$, $\angle \text{B}$, $\angle \text{C}$, and $\angle \text{D}$ each equal to $90^{\circ}$.

Examples of Rectangle

A rectangle is the most common shape that we see around ourselves in our daily lives. Some of the common examples of rectangular shapes are table top, blackboard, notebook, door, window, envelope, drawer, etc.

Other Names of Rectangle

Rectangles are a special case of parallelograms. These are the other names of a rectangle.

• A rectangle is also called an equiangular quadrilateral since all the angles of a rectangle are equal.
• It is also called a right-angled parallelogram since a rectangle has parallel sides.

Practice Problems

1. What is a rectangle?
2. Under what condition a parallelogram is called a rectangle?
3. Under what condition a rectangle is called a square?
4. State True or False
   • Adjacent sides of a rectangle are equal
   • Opposite sides of a rectangle are equal
   • Adjacent sides of a rectangle are parallel
   • Adjacent sides of a rectangle are perpendicular
   • Opposite sides of a rectangle are parallel
   • Opposite sides of a rectangle are perpendicular
   • Diagonals of a rectangle are equal
   • Diagonals of a rectangle are unequal
   • Diagonals of a rectangle are perpendicular to each other
   • Diagonals of a rectangle are parallel to each other

FAQs

Conclusion

A rectangle is a closed 2D shape, having four sides, four angles, and four vertices. Each of the four angles of a rectangle is a right angle, i.e., measure $90^{\circ}$. The opposite sides of a rectangle are equal and parallel. Since a rectangle is a 2D shape, it is characterized by two dimensions, a longer one is length, and a shorter one is width.

Recommended Reading

• What is a Rhombus – Definition, Types, Properties & Examples
• What is a Trapezium – Definition, Types, Properties & Examples
• What is a Parallelogram – Definition, Properties & Examples
• Types of Quadrilaterals and Their Properties(With Definitions & Examples)
• What is Quadrilateral in Math(Definition, Shape & Examples)
• Angle Bisector of a Triangle – Definition, Properties & Examples
• Similarity of Triangles Criteria – SSS, SAS, AA
• Congruence of Triangles Criteria – SSS, SAS, ASA, RHS
• Altitude of a Triangle(Definition & Properties)
• Median of a Triangle(Definition & Properties)
• How to Construct a Triangle(With Steps, Diagrams & Examples)
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• 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
• Types of Triangles – Definition & Examples
• What is Triangle in Geometry – Definition, Shapes & Examples
• Pair of Angles – Definition, Diagrams, Types, and Examples
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• Types of Angles in Maths(Acute, Right, Obtuse, Straight & Reflex)
• What is an Angle in Geometry – Definition, Properties & Measurement
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• 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
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• What is a Perpendicular Line in Geometry – Definition, Properties & Examples
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