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 2020-21 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 2020-21 136 MATHEMA TICS 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. 8.4 2020-21 Page 3 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 2020-21 136 MATHEMA TICS 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. 8.4 2020-21 QUADRILATERALS 137 You know that ? DAC + ? ACD + ? D = 180° (1) Similarly, in ? ABC, ? CAB + ? ACB + ? B = 180° (2) Adding (1) and (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°. 8.3 Types of Quadrilaterals Look at the different quadrilaterals drawn below: Fig. 8.5 Observe that : l One pair of opposite sides of quadrilateral ABCD in Fig. 8.5 (i) namely, AB and CD are parallel. You know that it is called a trapezium. l Both pairs of opposite sides of quadrilaterals given in Fig. 8.5 (ii), (iii) , (iv) and (v) are parallel. Recall that such quadrilaterals are called parallelograms. So, quadrilateral PQRS of Fig. 8.5 (ii) is a parallelogram. 2020-21 Page 4 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 2020-21 136 MATHEMA TICS 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. 8.4 2020-21 QUADRILATERALS 137 You know that ? DAC + ? ACD + ? D = 180° (1) Similarly, in ? ABC, ? CAB + ? ACB + ? B = 180° (2) Adding (1) and (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°. 8.3 Types of Quadrilaterals Look at the different quadrilaterals drawn below: Fig. 8.5 Observe that : l One pair of opposite sides of quadrilateral ABCD in Fig. 8.5 (i) namely, AB and CD are parallel. You know that it is called a trapezium. l Both pairs of opposite sides of quadrilaterals given in Fig. 8.5 (ii), (iii) , (iv) and (v) are parallel. Recall that such quadrilaterals are called parallelograms. So, quadrilateral PQRS of Fig. 8.5 (ii) is a parallelogram. 2020-21 138 MATHEMA TICS Similarly, all quadrilaterals given in Fig. 8.5 (iii), (iv) and (v) are parallelograms. l In parallelogram MNRS of Fig. 8.5 (iii), note that one of its angles namely ? M is a right angle. What is this special parallelogram called? Try to recall. It is called a rectangle. l The parallelogram DEFG of Fig. 8.5 (iv) has all sides equal and we know that it is called a rhombus. l The parallelogram ABCD of Fig. 8.5 (v) has ? A = 90° and all sides equal; it is called a square. l In quadrilateral ABCD of Fig. 8.5 (vi), AD = CD and AB = CB i.e., two pairs of adjacent sides are equal. It is not a parallelogram. It is called a kite. Note that a square, rectangle and rhombus are all parallelograms. l A square is a rectangle and also a rhombus. l A parallelogram is a trapezium. l A kite is not a parallelogram. l A trapezium is not a parallelogram (as only one pair of opposite sides is parallel in a trapezium and we require both pairs to be parallel in a parallelogram). l A rectangle or a rhombus is not a square. Look at the Fig. 8.6. We have a rectangle and a parallelogram with same perimeter 14 cm. Fig. 8.6 Here the area of the parallelogram is DP × AB and this is less than the area of the rectangle, i.e., AB × AD as DP < AD. Generally sweet shopkeepers cut ‘Burfis’ in the shape of a parallelogram to accomodate more pieces in the same tray (see the shape of the Burfi before you eat it next time!). Let us now review some properties of a parallelogram learnt in earlier classes. 2020-21 Page 5 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 2020-21 136 MATHEMA TICS 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. 8.4 2020-21 QUADRILATERALS 137 You know that ? DAC + ? ACD + ? D = 180° (1) Similarly, in ? ABC, ? CAB + ? ACB + ? B = 180° (2) Adding (1) and (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°. 8.3 Types of Quadrilaterals Look at the different quadrilaterals drawn below: Fig. 8.5 Observe that : l One pair of opposite sides of quadrilateral ABCD in Fig. 8.5 (i) namely, AB and CD are parallel. You know that it is called a trapezium. l Both pairs of opposite sides of quadrilaterals given in Fig. 8.5 (ii), (iii) , (iv) and (v) are parallel. Recall that such quadrilaterals are called parallelograms. So, quadrilateral PQRS of Fig. 8.5 (ii) is a parallelogram. 2020-21 138 MATHEMA TICS Similarly, all quadrilaterals given in Fig. 8.5 (iii), (iv) and (v) are parallelograms. l In parallelogram MNRS of Fig. 8.5 (iii), note that one of its angles namely ? M is a right angle. What is this special parallelogram called? Try to recall. It is called a rectangle. l The parallelogram DEFG of Fig. 8.5 (iv) has all sides equal and we know that it is called a rhombus. l The parallelogram ABCD of Fig. 8.5 (v) has ? A = 90° and all sides equal; it is called a square. l In quadrilateral ABCD of Fig. 8.5 (vi), AD = CD and AB = CB i.e., two pairs of adjacent sides are equal. It is not a parallelogram. It is called a kite. Note that a square, rectangle and rhombus are all parallelograms. l A square is a rectangle and also a rhombus. l A parallelogram is a trapezium. l A kite is not a parallelogram. l A trapezium is not a parallelogram (as only one pair of opposite sides is parallel in a trapezium and we require both pairs to be parallel in a parallelogram). l A rectangle or a rhombus is not a square. Look at the Fig. 8.6. We have a rectangle and a parallelogram with same perimeter 14 cm. Fig. 8.6 Here the area of the parallelogram is DP × AB and this is less than the area of the rectangle, i.e., AB × AD as DP < AD. Generally sweet shopkeepers cut ‘Burfis’ in the shape of a parallelogram to accomodate more pieces in the same tray (see the shape of the Burfi before you eat it next time!). Let us now review some properties of a parallelogram learnt in earlier classes. 2020-21 QUADRILATERALS 139 8.4 Properties of a Parallelogram Let us perform an activity. Cut out a parallelogram from a sheet of paper and cut it along a diagonal (see Fig. 8.7). You obtain two triangles. What can you say about these triangles? Place one triangle over the other. Turn one around, if necessary. What do you observe? Observe that the two triangles are congruent to each other. Repeat this activity with some more parallelograms. Each time you will observe that each diagonal divides the parallelogram into two congruent triangles. Let us now prove this result. Theorem 8.1 : A diagonal of a parallelogram divides it into two congruent triangles. Proof : Let ABCD be a parallelogram and AC be a diagonal (see Fig. 8.8). Observe that the diagonal AC divides parallelogram ABCD into two triangles, namely, ? ABC and ? CDA. We need to prove that these triangles are congruent. In ? ABC and ? CDA, note that BC || AD and AC is a transversal. So, ? BCA = ? DAC (Pair of alternate angles) Also, AB || DC and AC is a transversal. So, ? BAC = ? DCA (Pair of alternate angles) and AC = CA (Common) So, ? ABC ? ? CDA (ASA rule) or, diagonal AC divides parallelogram ABCD into two congruent triangles ABC and CDA. Now, measure the opposite sides of parallelogram ABCD. What do you observe? You will find that AB = DC and AD = BC. This is another property of a parallelogram stated below: Theorem 8.2 : In a parallelogram, opposite sides are equal. Y ou have already proved that a diagonal divides the parallelogram into two congruent Fig. 8.7 Fig. 8.8 2020-21Read More

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