17. Quadrilateral And Properties of Quadrilateral

 

QUADRILATERAL AND PROPERTIES OF QUADRILATERAL

Learn and Remember: .

 1. Quadrilateral :

(i) If A, B, C, D are coplanar points such that no three of them are collinear, then the union of the segments AB, BC, CD and DA is called the quadrilateral ABCD.

The quadrilateral ABCD (written as ☐ABCD) is shown in the figure.

A, B, C, D are called the vertices of the quadrilateral.

(ii) Seg AB, seg BC, seg CD and seg DA are the four sides of ☐ABCD.

(iii) Seg AB and seg BC, seg BC and seg CD, seg CD and seg DA, seg DA and seg AB are the four pairs of adjacent sides. The two adjacent sides have a common vertex.

(iv) Seg AB and seg DC; seg BC and seg AD are the two pairs of opposite sides. The opposite sides have no vertex in common.

(v) ∠A and ∠C; ∠B and ∠D are the two pairs of opposite angles.

(vi) ∠A and ∠B; ∠B and ∠C; ∠C and ∠D; ∠D and ∠A are the four pairs of adjacent angles. The adjacent angles have a common arm. •  The sum of the measures of all the angles of a quadrilateral is 360°.

•   If the sides of a quadrilateral are produced in order,  the sum of the measures of the exterior angles so formed is equal to 360°. In the figure,

m∠1+ m∠2 + m ∠3+m∠4=360°

2. Types of Quadrilaterals :

(i) Parallelogram : If both the pairs of opposite sides of a quadri lateral are parallel, the quadrilateral is called a parallelogram, In the figure, ☐ABCD has side AD || side BC and side AB | side DC. 

∴ ☐ABCD is a parallelogram.

Properties of a parallelogram :

(a) Opposite sides of a parallelogram are congruent.

e.g. in the given figure, side AD ≅ side BC and side AB ≅ side DC.

(b) Opposite angles of a parallelogram are congruent.

In the given figure, ∠A ≅∠C and ∠B ≅ ∠D.

(c) Diagonals of a parallelogram bisect each other.

In the given figure, diagonals AC and BD bisect each other at O.

i.e. seg AO ≅seg CO and seg BO ≅ seg DO.

(ii) Rhombus : A parallelogram having all its sides congruent is called a rhombus.

In the figure, the parallelogram ABCD has side AB ≅side BC ≅ side CD≅side DA. .

☐ABCD is a rhombus.

Properties of a rhombus :

(a) Since a rhombus is a parallelogram, all the properties of a parallelogram hold good for a rhombus.

(b) Diagonals of a rhombus bisect each other at right angles. In the given figure, diagonals AC and BD of the rhombus ABCD, are perpendicular to each other. Also seg AO ≅ seg CO and seg BO ≅ seg DO.

(c) Diagonals of a rhombus bisect its angles.

(iii) Rectangle : A parallelogram having each of its angles a right angle, is called a rectangle.

☐ABCD is a parallelogram with each of the angles, ∠A, ∠B, ∠C and ∠D, a right angle.

∴ ☐ ABCD is a rectangle.

Properties of a rectangle :

(a) Since a rectangle is a parallelogram, all the properties of a parallelogram hold good for a rectangle.

(b) Diagonals of a rectangle are congruent. In the given figure, rectangle ABCD has diagonal AC ≅ diagonal BD.

(iv) Square : When all the angles of a rhombus are right angles, it is called a square. In the figure, all the angles of the rhombus ABCD are right angles. ☐ABCD is a square.

Properties of a square : All the properties of a rhombus and a rectangle hold good for a square. To summarise :

(a) All the sides of a square are congruent. Thus, side AB ≅ side BC≅ side CD ≅ side DA.

(b) All the angles of a square are right angles. Thus, M∠A = m ∠B = m∠C = m∠D = 90°.

(c) Diagonals of a square are congruent. Thus, diagonal AC ≅ diagonal BD.

(d) Diagonals of a square are perpendicular bisectors of each other. Thus, diagonal

AC ⊥ diagonal BD. Also, seg AO ≅ seg CO ≅seg BO ≅seg DO.

(v) Trapezium : A quadrilateral having only one pair of opposite sides parallel, is called a trapezium. In the figure, ☐ABCD is a trapezium having side AD || side BC.

(vi) Isosceles trapezium : If the non-parallel sides of a trapezium are congruent, then it is called an isosceles trapezium. In the figure, the trapezium ABCD has side

 AB ≅ side DC.

☐ABCD is an isosceles trapezium.

Properties of an isosceles trapezium:

(a)  Diagonals of an isosceles trapezium are congruent. Thus, in the given figure of the isosceles trapezium ABCD, diagonal AC ≅ diagonal BD.

(b)  The pairs of angles at the end of the parallel sides of an isosceles trapezium are congruent. Thus, in the given figure, ∠A = ∠D and ∠B = ∠C.

(c) Opposite angles of an isosceles trapezium are supplementary. Thus, in the given figure, m∠A+m∠C= 180°

and m∠B + m∠ D = 180°. 

(vi) Kite: In the figure, ☐ABCD is a kite. Side AB ≅ side AD.

side BC ≅ side DC but side AD≅ side DC.

Properties of a kite :

(a) One pair of opposite angles of a kite are congruent. In the adjoining figure of the kite ABCD,

∠ABC = ∠LADC.

(h) The diagonal that joins the points of intersection of the congruent sides of the kite bisects the other diagonal at right angles. Diagonal AC bisects diagonal BD at right angles.