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A four-sided closed two-dimensional shape is called a quadrilateral. It has four vertices, four sides and four angles.
Angle sum property of a quadrilateral
The sum of the angles of a quadrilateral is 360º.
Proof:
Given: ABCD is a quadrilateral.
Construction: Join BD.

Now, we divided the quadrilateral into two triangles BAD and BCD.
\angle B = \angle ABD + \angle DBC - - - - - (I)
\angle D = \angle ADB + \angle BDC - - - - - (II)
We know that "sum of all the angles of a triangle is 180^\circ".
In \Delta BAD:
\angle DAB + \angle ABD + \angle BDA = 180^\circ - - - - (III)
Similarly, in \Delta BCD,
\angle DBC + \angle BCD + \angle CDB = 180^\circ - - - - (IV)
Adding equations (III) and (IV), we get:
\angle DAB + \angle ABD + \angle BDA + \angle DBC + \angle BCD + \angle CDB = 180^\circ + 180^\circ
Rearrange the angles.
\angle DAB + (\angle ABD + \angle DBC) + \angle BCD + (\angle BDA + \angle CDB) = 360^\circ
\angle A + \angle B + \angle C + \angle D = 360^\circ [using equations (I) and (II)]
That is, the sum of the angles of a quadrilateral is 360^\circ.
Types of quadrilateral
Name | Picture | Properties |
Parallelogram | ![]() | 1. Opposite sides are equal and parallel. 2. Opposite angles are equal. 3. Diagonals bisect each other. |
Square | ![]() | 1. All sides are equal and parallel. 2. All interior angles are 90^\circ. 3. Diagonals bisect each other at right angles. |
Rectangle | ![]() | 1. Opposite sides are equal and parallel. 2. All interior angles are 90^\circ. 3. Diagonals bisect each other. |
Rhombus | ![]() | 1. All sides are equal. 2. Opposite angles are equal. 3. Diagonals are perpendicular. |
Trapezium | ![]() | 1. The bases of a trapezium are parallel. 2. Sum of adjacent angles on non-parallel sides are supplementary. |
Kite | ![]() | 1. Diagonals are perpendicular. 2. Diagonals bisect the vertex angles. 3. Non-vertex angles are congruent. 4. Two disjoint pairs of consecutive sides are congruent. |
Properties of a quadrilateral
1. A square, rectangle and rhombus are all parallelograms.
2. A square is a rectangle and also a rhombus.
3. A rectangle or a rhombus is not a square.
4. A parallelogram is a trapezium, but a trapezium is not a parallelogram.
5. A kite is not a parallelogram.