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**Geometry Notes on Quadrilateral Tips and Tricks**

In this article we will discuss about the quadrilateral which is an important topic of Geometry. This topic is useful both in **Geometry and Mensuration**. Here we cover all the quadrilateral and various properties and theorem related to them.

### Quadrilateral:

A plane figure bounded by four line segment is called the quadrilateral. The area enclosed by the four side must not be zero.

A simple quadrilateral has four sides and four angle.

Sum of the angle of quadrilateral is always** 360°**

**∠A+∠B+∠C+∠D=360°**

**Area of quadrilateral ABCD = 1/2[AC(DE+BF)]**

DE and BF are the perpendicular on diagonal AC.

### Types of Quadrilateral

**Parallelogram:**

- A quadrilateral whose opposite sides are
**equal and parallel** - Opposite sides angle are equal
**(∠A=∠C)and (∠B=∠D).** - Sum of any adjacent angle is
**180°** - Diagonals
**bisect**each other. - Diagonals need not be perpendicular or equal.
- Each diagonals divides a parallelogram into two
**triangle ofequal area**. - Area of parallelogram is double the area of the triangle formed by diagonals.
- Bisectors of the angle of a parallelogram form a
**rectangle**.

- A parallelogram inscribed inside a circle is
**rectangle**. - A parallelogram circumscribed about a circle is a
**rhombus.** **AC**^{2}+BD^{2}=AB^{2}+BC^{2}+CD^{2}+AD^{2}=2(AB^{2}+BC^{2})- Area of parallelogram
**ABCD= Base *Height** - A parallelogram is a rectangle if its diagonal are equal.
- Parallelogram that lie on the same base and between the same parallel lines are equal in area.

**area of Parallelogram ABCD=Area of Parallelogram ABPQ**

- If a triangle and a parallelogram are on same base and between the same parallel lines , then area of triangle is half of parallelogram.

**Area of triangle APB=1/2(area of parallelogram ABCD)**

**Rectangles:**

- A rectangle is a parallelogram whose all angles
**90°.** - Diagonal are equal and bisect each other but not necessarily at
**90°**. - A square has
**maximum area**for a given perimeter of rectangles. - Figures formed by joining the mid points of the adjacent side of the rectangle is a
**rhombus.** - Angle
**Bisectors**of a rectangle form another**rectangle.** **Area of rectangle=length*breadth****Diagonals of the rectnagle[AC=BD=√(l**^{2}+b^{2})]

**Square:**

- A square is a rectangle with all its
**side equa**l and all the angle equal to 90°. - Diagonals are equal and bisect each other at 90°
- Figure formed by joining the midpoint of the sides of square is a square.
**Area= a**^{2}**Diagonal=√2*a**

**Rhombus:**

- A parallelogram having
**all its sides equal**is a rhombus. - Diagonals of the rhombus
**bisect**each other at right angle but they are not necessarily equal. - A rhombus may or may not be a square but all the square are rhombus.
- Figure formed by joining the midpoint of the adjacent sides of a rhombus is a rectangle.
- A parallelogram is a rhombus if its diagonals are perpendicular to each other.
**Area of rhombus=1/2(product of diagonal)**- here AC and BD are diagonal.
**AC**^{2}+BD^{2}=4AB^{2}

**Trapezium:**

- A quadrilateral is a trapezium with only two sides parallel to each other.

- sum of opposite angle equal to 180(∠A+∠C=∠B+∠D=180°)
- Area of trapezium =1/2[(sum of parallel sides)*heights]

** = 1/2[(AB+CD)*DM]**

- sum of the square of diagonal =(sum of square of non parallel sides) +2(products of parallel sides)

**AC ^{2}+BD^{2}=BC^{2}+AD^{2}+2AB*CD**

- If E and F are the midpoints of two non-parallel sides AD and BC respectively then

**EF=1/2(AB+BC) **

**AO*OD=OC*BO**

- E and F are the mid-points of the side AB and AC respectively then,

**EF=1/2(BC) and EF parllel BC**

**Polygon:**

- A closed figure bounded by three or more than three straight lines.

**Convex Polygon: **

- A polygon in which none of its interior angle is more than 180°.

**Concave Polygon:**

- A polygon in which at-least one of its interior angle is more than 180°

**Regular Polygon:**

- A polygon in which all the sides are equal and also the interior angles are equal is called a regular polygon.
- if n is the total no sides of the polygon then
- Sum of its interior angle is
**(n-2)*180°** - Each exterior angle=
**360°/n** - Sum of all exterior angle is 360°
- Sum of interior angle and exterior angle is 180°
- Number of diagonals =
**n(n-3)/2**