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INTRODUCTION
We have studied in detail about the properties of a triangle. We also know that the triangle is a figure obtained by joining three non collinear points in pair. In this chapter we shall discuss about four noncollinear points such that no three of them are collinear.
1. QUADRILATERALS
We know that the figure obtained on joining three noncollinear points in pairs is a triangle. If we mark four points and join them in some order, then there are three possibilities for the figure obtained:
(i) If all the points are collinear (in the same line), we obtain a line segment.
(ii) If three out of four points are collinear, we get a triangle.
(iii) If no three points out of four are collinear, we obtain a closed figure with four sides.
Each of the figure obtained by joining four points in order is called a quadrilateral. (quad means four and lateral for sides).
2. CONSTITUENTS OF A QUADRILATERAL
A quadrilateral has four sides, four angles and four vertices.
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. If we join the opposite vertices A to C and B to D, then AC and BD are the two diagonals of the quadrilateral ABCD.
3. QUADRILATERALS IN PRACTICAL LIFE
We find so many objects around us which are of the shape of a quadrilateral the floor, walls, ceiling, windows of our classroom, the blackboard, each face of the duster, each page of our mathematics book, the top of our study table, etc. Some of these are given below.
4. SOME RELATED TERMS TO QUADRILATERALS
In a quadrilateral ABCD, we have
(i) VERTICES : The points A, B, C and D are called the vertices of quadrilateral ABCD.
(ii) SIDES : The line segments AB, BC, CD and DA are called the sides of quadrilateral ABCD.
(iii) DIAGONALS : The line segments AC and BD are called the diagonals of quadrilateral ABCD.
(iv) ADJACENT SIDES : The sides of a quadrilateral are said to be adjacent sides if they have a common end point. Here, in the above figure, (AB,BC),(BC,CD),(CD,DA)and(DA,AB) are four pairs of adjacent sides or consecutive sides of quadrilateral ABCD.
(v) OPPOSITE SIDES : Two sides of a quadrilateral are said to be opposite sides if they have no common end point. Here, in the above figure, (AB, DC) and (BC, AD) are two pairs of opposite sides of quadrilateral ABCD.
(vi) CONSECUTIVE ANGLES : Two angles of a quadrilateral are said to be consecutive angles if they have a common arm. Here, in the above figure, (∠A,∠B), (∠B,∠C), (∠C, ∠D) and (∠D, ∠A) are four pairs of consecutive angles.
(vii) OPPOSITE ANGLES : Two angles of a quadrilateral are said to be opposite angles if they have no common arm. Here, in the given figure, (∠A, ∠C) and (∠B, ∠D) are two pairs of opposite angles of quadrilateral ABCD.
TYPES OF QUADRILATERALS
(i) PARALLELOGRAM : A quadrilateral in which Dboth pair of opposite Csides are parallel is called a parallelogram.
In figure, ABCD is a quadrilateral in which AB ⊥ DC, BC⊥ AD.
∴ quadrilateral ABCD is a parallelogram.
(ii) RHOMBUS : A parallelogram whose all sides are equal is called rhombus.
In figure, ABCD is a parallelogram in which AB = BC = CD = DA, AB ⊥ DC and BC ⊥ AD .
∴ parallelogram ABCD is a rhombus.
(iii) RECTANGLE : A parallelogram whose each angle is equal to 90°, is called a rectangle. In figure, ABCD is a parallelogram in which
∠A = ∠B = ∠C = ∠D = 90° ,
AB ⊥ DC and BC ⊥ AD.
∴ Parallelogram ABCD is a rectangle. A B
(iv) SQUARE : A rectangle in which a pair of adjacent sides are equal is said to be a square.
In figure, ABCD is a rectangle in which ∠A = ∠B = ∠C = ∠D = 90° ,
AB = BC, BC = CD, CD = DA, DA= AB.
i.e., AB = BC = CD = DA .
∴ rectangle ABCD is a square. A B
(v) TRAPEZIUM : A quadrilateral in which exactly one pair of opposite sides is parallel, is called a trapezium. D
A In figure, ABCD is a quadrilateral in which AB ⊥ DC.
∴ ABCD is trapezium.
(vi) ISOSCALES TRAPEZIUM : A trapezium whose nonparallel sides are equal is called an isosceles trapezium. D
A In figure, ABCD is a trapezium in which AB ⊥ DC and BC = AD .
∴ trapezium ABCD is isosceles trapezium.
(vii) KITE : A quadrilateral in which two pairs of adjacent sides are equal is called a kite. In figure, ABCD is a quadrilateral in which AB = AD and BC = CD.
∴ quadrilateral ABCD is a kite.
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