Quadrilaterals (original) (raw)

Last Updated : 30 Jan, 2026

A Quadrilateral is a two-dimensional closed shape formed by four straight line segments. It has four sides, four vertices (corners), and four interior angles. The sum of all interior angles of a quadrilateral is always 360°.

Quadrilaterals are either simplex (not self-intersecting), or complex (self-intersecting). Simple quadrilaterals are further divided into concave and convex.

Properties of a Quadrilateral

• It has four sides: AB, BC, CD, and DA.
• It has four vertices: A, B, C, and D.
• It has four angles: ∠DAB, ∠ABC, ∠BCD, and ∠CDA.
• ∠A and ∠B are adjacent angles.
• ∠A and ∠C are opposite angles.
• AB and CD are opposite sides.
• AB and BC are adjacent sides.

Simple Quadrilateral

A simple quadrilateral is a quadrilateral in which the sides do not intersect each other. It is a closed figure made of four straight line segments.

**Convex Quadrilaterals

A convex quadrilateral is a four-sided shape in which all interior angles are less than 180°, so the shape does not bend inward, and both diagonals lie completely inside the figure.

• **Irregular Quadrilateral: A quadrilateral with four sides that are unequal in length and angles.
• **Trapezium (Trapezoid): A quadrilateral with one pair of opposite sides parallel.
• **Isosceles Trapezium (Isosceles Trapezoid): A trapezium whose non-parallel sides are equal and whose base angles are equal.
• **Parallelogram: A quadrilateral with both pairs of opposite sides parallel.
• **Kite: A quadrilateral with two pairs of adjacent sides equal.
• **Rhombus: A quadrilateral with all four sides equal.
• **Rectangle: A quadrilateral with all interior angles equal to 90°.
• **Square: A quadrilateral with all sides equal and all angles equal to 90°.

**Concave Quadrilateral

A concave quadrilateral is a quadrilateral in which at least one interior angle is greater than 180°. Because of this, one of its diagonals lies partially or completely outside the figure.

Complex Quadrilateral

A complex quadrilateral is a quadrilateral in which the sides intersect each other, so the figure is self-crossing instead of simple.

Quadrilateral Theorems

• **Sum of Interior Angles Theorem: In any quadrilateral, the sum of all four interior angles is always 360 degrees. This is true for all four-sided shapes, whether regular or irregular.
• **Opposite Angles Theorem: In a quadrilateral, the sum of two opposite angles is 180 degrees. For example, in a parallelogram, each pair of opposite angles adds up to 180°.
• **Consecutive Angles Theorem: Adjacent (consecutive) angles in a quadrilateral are supplementary, meaning their sum is 180 degrees. This is particularly useful for parallelograms and trapeziums.
• **Diagonals of Parallelograms Theorem: The diagonals of a parallelogram bisect each other, dividing each diagonal into two equal parts. This property helps in proving congruence in triangles inside the parallelogram.
• **Opposite Sides and Angles of Parallelograms Theorem: In a parallelogram, opposite sides are equal in length and opposite angles are equal. This is why squares, rectangles, and rhombuses share this property.
• **Diagonals of Rectangles and Rhombuses Theorem: In a rectangle, the diagonals are equal in length. In a rhombus, the diagonals bisect each other at right angles. In both shapes, the diagonals divide the shape into congruent triangles.
• **Diagonals of Trapezoids Theorem: The diagonals of a trapezoid may be of different lengths. However, the line joining the midpoints of the non-parallel sides is always parallel to the bases and its length is half the sum of the parallel sides.

Quadrilateral Lines of Symmetry

A line of symmetry is an imaginary line that divides a shape into two identical halves, so that one half is a mirror image of the other.

**In quadrilaterals:

• A line of symmetry can pass through vertices or midpoints of sides.
• When folded along the line, the two halves match exactly.

**Examples:

• A square has four lines of symmetry: two along the diagonals and two along the lines joining the midpoints of opposite sides.
• A rectangle has two lines of symmetry, both joining the midpoints of opposite sides.
• Other quadrilaterals may have no lines of symmetry or fewer lines depending on their shape.

Learn in Detail

• Construction of a Quadrilateral
• Types and their Properties
• Formulas
• Area and Perimeter
• Real-Life Applications
• Practice Questions

Solved Examples on Quadrilaterals

**Question 1: The perimeter of quadrilateral ABCD is 46 units. AB = x + 7, BC = 2x + 3, CD = 3x - 8, and DA = 4x - 6. Find the length of the shortest side of the quadrilateral.

**Solution:

Perimeter = Sum of all sides

= 46 = 10x - 4 or [x = 5]

That gives, AB = 12 units, BC = 13 units, CD = 7 units, DC = 14 units

Hence, l**ength of shortest side is 7 units (i.e. CD).

**Question 2: Given a trapezoid ABCD (AB || DC) with median EF. AB = 3x - 5, CD = 2x -1 and EF = 2x + 1. Find the value of EF.

**Solution:

We know that the Median of the trapezoid is half the sum of its bases.

= EF = (AB + CD) / 2

= 4x + 2 = 5x - 6 or [x = 8]

Therefore EF = 2x + 1 = 2(8) + 1 => EF = 17 units.

**Question 3: In a Parallelogram, adjacent angles are in the ratio of 1:2. Find the measures of all angles of this Parallelogram.

**Solution:

Let the adjacent angle be x and 2x.

We know that in of a Parallelogram adjacent angles are supplementary.

= x + 2x = 180° or [x = 60°]

Also, opposite angles are equal in a Parallelogram.

Therefore measures of each angles are **60°, 120°, 60°, 120°.