# Important Geometry Notes on Quadrilateral

By Sachin Awasthi|Updated : February 10th, 2021

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

A simple quadrilateral has four sides and four angles.

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

A simple quadrilateral has four sides and four angles. Sum of the angle of the quadrilateral is always 360°

∠A+∠B+∠C+∠D=360°

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

DE and BF are the perpendiculars 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 a 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.
• Area of parallelogram ABCD= Base *Height
• A parallelogram is a rectangle if its diagonal are equal.
• Parallelogram that lies 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 the same base and between the same parallel lines, the area of triangle is half of the parallelogram. Area of triangle APB=1/2(area of parallelogram ABCD)

Rectangles: • A rectangle is a parallelogram whose all angles 90°.
• Diagonals 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 midpoints 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=√(l2+b2)]

Square: • A square is a rectangle with all its side equal 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= a2
• Diagonal=√2*a

Rhombus: • A parallelogram having all its sides equal is a rhombus.
• Diagonals of the rhombus bisect each other at a right angle but they are not necessarily equal.
• A rhombus may or may not be a square but all the square are rhombus.
• The 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.
• AC2+BD2=4AB2

Trapezium:

• A quadrilateral is a trapezium with only two sides parallel to each other. • The sum of co-interior angles are equal to 180(∠A+∠D=180°, B +C=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)

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

EF=1/2(AB+DC) • 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 an interior angle and exterior angle is 180°
• Number of diagonals =n(n-3)/2

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