**Curve:** A figure formed on a plane surface by joining a number of points without lifting a pencil is called a curve.

**Open Curve:** A curve which does not end at the same starting point or which does not cut itself is called an open curve.

**Closed Curve:** A curve which cut itself or which starts and ends at the same point is called a closed curve.

**Simple Closed Curve:** A closed curve called a simple closed curve which does not intersect itself.

**Polygon:** A polygon is a closed figure bounded by three or more line segments such that each line segment intersects exactly two other points (vertices) as shown in the following figures.

**Quadrilateral:** A simple closed figure bounded by four line segments is called a quadrilateral, it has four sides i.e., AB, BC, CD and AD and four vertices as A, B, C and D and the sum of all angles of a quadrilateral is 360°.

**Parallelogram:** A quadrilateral in which opposite sides are parallel and equal is called parallelogram; written as || gm. and the diagonals of a parallelogram bisect each other.

**Properties:**

- Opposite sides are equal and parallel.
- Opposite angles are equal.
- Diagonals bisect each other.

**Rectangle:** A parallelogram each of whose angle is 90° and diagonals are equal, is called a rectangle.

**Properties:**

- Opposite sides are equal and parallel.
- Each angle is a right angle.
- Diagonals are equal.
- Diagonals bisect each other.

**Square:** A quadrilateral in which all sides and angles are equal, is called a square.

**Properties:**

- All the sides are equal and parallel.
- Each angle is a right angle.
- Diagonals are equal.
- Diagonals bisect each other at a right angle.

**Rhombus:** A parallelogram having all its sides equal, is called a rhombus.

**Properties:**

- All the side are equal.
- Opposite angles are equal.
- Diagonals bisect each other at a right angle.

**Trapezium:** A quadrilateral in which two opposite sides are parallel and the other two opposite sides are non-parallel, is called a trapezium.

If two non-parallel sides of a trapezium are equal, then it is called an isosceles trapezium.

The line segment joining the mid-points of non-parallel sides of a trapezium is called it’s median.

**Kite:** A quadrilateral in which two pairs of adjacent sides are equal, is called a kite.

**Properties:**

Diagonals bisect each other at the right angle.

In the figure, m ∠B = m ∠D, but m∠A ≠ m ∠C

Paper is a very common example of a plane surface. The curve obtained by joining a number of points consecutively without lifting the pencil from the paper is called a plane curve. A circle is a very common example of a plane curve.

A polygon is a simple closed curve formed of only line segments. A triangle is a very common example of a polygon.

**Classification of Polygons**

A polygon is said to be a triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon, ………, n-gon according as its number of sides (or vertices) is 3, 4, 5, 6, 7, 8, 9, 10, ……… , n respectively.

Diagonals

The line-segment joining any two non-consecutive vertices of a polygon is called its diagonal.

Convex and Concave Polygons

A polygon is said to be convex if it has no portion of its diagonals in its exterior otherwise it is said to be a concave polygon.

Regular and Irregular Polygons

A polygon which is both ‘equiangular’ (has all angles of equal measure) and ‘equilateral’ (has all sides of equal measure) is called a regular polygon, for example, a square, an equilateral triangle.

A polygon which is equiangular but not equilateral is called an irregular polygon. For example; a rectangle.

The sum of the measures of the three angles of a triangle is 180°.

The sum of the measures of the exterior angles of a polygon is 360°.

Kinds of Quadrilaterals

The important types of quadrilaterals are as follows:

- Trapezium
- Kite
- Parallelogram
- Rhombus
- Rectangle
- Square.

Trapezium

A quadrilateral which has only one pair of parallel sides is called a trapezium.

Kite

A quadrilateral, which has exactly two pairs of equal consecutive sides, is called a kite.

Parallelogram

A quadrilateral whose opposite sides are parallel is called a parallelogram.

Elements of a Parallelogram

The elements of a parallelogram are as follows:

- two pairs of opposite sides
- four pairs of adjacent sides
- two pairs of opposite equal angles
- four pairs of adjacent angles.

The opposite sides of a parallelogram are of equal length.

The opposite angles of a parallelogram are of equal measure.

The adjacent angles in a parallelogram are supplementary.

The diagonals of a parallelogram bisect each other at their point of intersection.

Rhombus

A quadrilateral whose all the four sides are of equal length is called a rhombus.

The diagonals of a rhombus are perpendicular bisectors of each other.

Rectangle

A rectangle is a parallelogram with equal angles.

The diagonals of a rectangle are of equal length.

Square

A square is a rectangle whose all the four sides are equal.

The diagonals of a square are perpendicular bisectors of each other.