Page 1 CHAPTER 8 QUADRILATERALS 8.1 Introduction You have studied many properties of a triangle in Chapters 6 and 7 and you know that on joining three non-collinear points in pairs, the figure so obtained is a triangle. Now, let us mark four points and see what we obtain on joining them in pairs in some order. Fig. 8.1 Note that if all the points are collinear (in the same line), we obtain a line segment [see Fig. 8.1 (i)], if three out of four points are collinear, we get a triangle [see Fig. 8.1 (ii)], and if no three points out of four are collinear, we obtain a closed figure with four sides [see Fig. 8.1 (iii) and (iv)]. Such a figure formed by joining four points in an order is called a quadrilateral. In this book, we will consider only quadrilaterals of the type given in Fig. 8.1 (iii) but not as given in Fig. 8.1 (iv). A quadrilateral has four sides, four angles and four vertices [see Fig. 8.2 (i)]. Fig. 8.2 Page 2 CHAPTER 8 QUADRILATERALS 8.1 Introduction You have studied many properties of a triangle in Chapters 6 and 7 and you know that on joining three non-collinear points in pairs, the figure so obtained is a triangle. Now, let us mark four points and see what we obtain on joining them in pairs in some order. Fig. 8.1 Note that if all the points are collinear (in the same line), we obtain a line segment [see Fig. 8.1 (i)], if three out of four points are collinear, we get a triangle [see Fig. 8.1 (ii)], and if no three points out of four are collinear, we obtain a closed figure with four sides [see Fig. 8.1 (iii) and (iv)]. Such a figure formed by joining four points in an order is called a quadrilateral. In this book, we will consider only quadrilaterals of the type given in Fig. 8.1 (iii) but not as given in Fig. 8.1 (iv). A quadrilateral has four sides, four angles and four vertices [see Fig. 8.2 (i)]. Fig. 8.2 136 MATHEMA TICS File Name : C:\Computer Station\Maths-IX\Chapter\Chap-6\Chap-6 (02-01-2006).PM65 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 [see Fig. 8.2 (ii)]. AC and BD are the two diagonals of the quadrilateral ABCD. In this chapter, we will study more about different types of quadrilaterals, their properties and especially those of parallelograms. You may wonder why should we study about quadrilaterals (or parallelograms) Look around you and you will find so many objects which are of the shape of a quadrilateral - the floor, walls, ceiling, windows of your classroom, the blackboard, each face of the duster, each page of your book, the top of your study table etc. Some of these are given below (see Fig. 8.3). Fig. 8.3 Although most of the objects we see around are of the shape of special quadrilateral called rectangle, we shall study more about quadrilaterals and especially parallelograms because a rectangle is also a parallelogram and all properties of a parallelogram are true for a rectangle as well. 8.2 Angle Sum Property of a Quadrilateral Let us now recall the angle sum property of a quadrilateral. The sum of the angles of a quadrilateral is 360º. This can be verified by drawing a diagonal and dividing the quadrilateral into two triangles. Let ABCD be a quadrilateral and AC be a diagonal (see Fig. 8.4). What is the sum of angles in ? ADC? Fig. 8.4Read More

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