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TOPICS
Introduction

A figure formed by joining four points in an order is called a quadrilateral. And a quadrilateral has four sides, four angles and four vertices. Now, In quadrilateral ABCD, AB, BC, CD and DA are the four sides; A, B, C and D are the four vertices and ∠ A, ∠ B, ∠ C and ∠ D are the four angles formed at the vertices.
Now join the opposite vertices A to C and B to D. AC and BD are the two diagonals of the quadrilateral ABCD.

Angle Sum Property of a Quadrilateral

To Prove : The sum of the angles of a quadrilateral is 360º. Proof : Let ABCD be a quadrilateral and AC be a diagonal.
We know that ∠ DAC + ∠ ACD + ∠ D = 180°.....(1)
Similarly, in Δ ABC, ∠ CAB + ∠ ACB + ∠ B = 180°.....(2)
We get ∠ DAC + ∠ ACD + ∠ D + ∠ CAB + ∠ ACB + ∠ B = 180° + 180° = 360°
Also, ∠ DAC + ∠ CAB = ∠ A and ∠ ACD + ∠ ACB = ∠ C
So, ∠ A + ∠ D + ∠ B + ∠ C = 360°.
i.e., the sum of the angles of a quadrilateral is 360°.

There are six types of Quadrilaterals arefollowing below :
(i) Trapezium
(ii) Parallelogram
(iii) Rectangle
(iv) Rhombus
(v) Square
(vi) Kite

(i) Trapezium

In Quadrilateral ABCD, One pair of opposite sides AB and CD are parallel. It is called a trapezium. Properties :
1. Two opposite sides are parallel.
2. Another two sides are non-parallel.
3. When two opposite sides are equal and another two sides are equal but not parallel then diagonals are equal.
4. When two opposite sides are equal and another two sides are not equal as well as not parallel then diagonals are not equal.

(ii) Parallelogram

In Quadrilateral PQRS, Both pairs of opposite sides of quadrilaterals are parallel. It is called a Parallelogram. Properties :
1. Opposite sides are parallel.
2. Opposite sides are equal.
3. When all sides are equal and opposite sides are parallel to each other then diagonals are equal.
4. A diagonal of a parallelogram divides it into two congruent triangles.
5. The diagonals of a parallelogram bisect each other.

(iii) Rectangle

In Quadrilateral MNRS, one of its ∠M is right-angle. It is called a Rectangle. Properties :
1. One of its angle is 90 degree.
2. Opposite sides are equal as well as parallel.
3. When all sides are equal and opposite sides are parallel to each other then diagonals are equal.
4. It is also a parallelogram.
5. Diagonals of a rectangle bisect each other and are equal and vice-versa.

(iv) Rhombus

In Quadrilateral DEFG, all sides are equal and parallel. It is called a Rhombus. Properties :
1. All sides are equal and pair of opposite sides are parallel.
2. It is also a parallelogram.
3. Diagonals of a rhombus bisect each other at right angles and vice-versa

(v) Square

In Quadrilateral ABCD, Both pairs of opposite sides of quadrilaterals are parallel. It is called a Square. Properties :
1. Pair of Opposite sides are parallel.
2. All sides are equal.
3. It is also a paralleogram, Rectangle and Rhombus.
4. Diagonals of a square bisect each other at right angles and are equal, and vice-versa.

(vi) Kite

In quadrilateral ABCD, AD = CD and AB = CB i.e., two pairs of adjacent sides are equal. It is not a parallelogram. It is called a kite. Properties :
1. Two pair of Adjacent sides are equal only.
2. Diagonals cut on 90 degree angle.

The Mid-point Theorem

The line-segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half of it.
A line through the mid-point of a side of a triangle parallel to another side bisects the third side.
The quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order, is a parallelogram. Properties :
1. EF = BC/2
2. ∠ AEF = ∠ ABC
3. so, EF || BC

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