Page 1 Points to Remember : 1. Quadrilateral. A simple closed figure bounded by four line segments is called a quadrilateral. In the adjacent figure, ABCD is a quadrilateral. A quadrilateral ABCD has : (i) Four vertices, namely A, B, C and D. (ii) Four sides, namely AB, BC, CD and DA. (iii) Four angles, namely DAB, ABC, BCD and CDA, to be denoted by A, B, C and D respectively. (iv) Two diagonals, namely AC and BD. (i) Adjacent sides. Two sides of a quadrilateral which have a common end point are called its adjacent sides. Thus AB, BC ; BC, CD ; CD, DA and DA, AB are four pairs of adjacent sides of the quadrilateral ABCD. (ii) Opposite sides. Two sides of a quadrilateral are called its opposite sides if they do not have a common end point. Thus AB, DC and AD, BC are two pairs of the opposite sides of the quadrilateral ABCD. (iii) Adjacent angles. Two angles of a quadrilateral having one common side are called its adjacent angles. Thus A, B ; B, C ; C, D and D, A are four pairs of adjacent angles of the quadrilateral ABCD. (iv) Opposite angles. Two angles of a quadrilateral which are not adjacent angles are known as the opposite angles of the quadrilateral. Thus A, C, and B, D are two pairs of opposite angles of the quadrilateral ABCD. 2. Convex and concave quadrilaterals (i) Convex quadrilateral. A quadrilateral in which the measure of each angle is less than 180º is called a convex quadrilateral. (ii) Concave quadrilateral. A quadrilateral in which the measure of at least one of the angles is more than 180º is called a concave quadrilateral. In the above figure, PQRS is a concave quadrilateral in which S > 180º. In this chapter, by a quadrilateral, we would mean a convex quadrilateral. 3. Interior and exterior of a quadrilateral Consider a quadrilateral ABCD. It divides the whole plane into three parts. Page 2 Points to Remember : 1. Quadrilateral. A simple closed figure bounded by four line segments is called a quadrilateral. In the adjacent figure, ABCD is a quadrilateral. A quadrilateral ABCD has : (i) Four vertices, namely A, B, C and D. (ii) Four sides, namely AB, BC, CD and DA. (iii) Four angles, namely DAB, ABC, BCD and CDA, to be denoted by A, B, C and D respectively. (iv) Two diagonals, namely AC and BD. (i) Adjacent sides. Two sides of a quadrilateral which have a common end point are called its adjacent sides. Thus AB, BC ; BC, CD ; CD, DA and DA, AB are four pairs of adjacent sides of the quadrilateral ABCD. (ii) Opposite sides. Two sides of a quadrilateral are called its opposite sides if they do not have a common end point. Thus AB, DC and AD, BC are two pairs of the opposite sides of the quadrilateral ABCD. (iii) Adjacent angles. Two angles of a quadrilateral having one common side are called its adjacent angles. Thus A, B ; B, C ; C, D and D, A are four pairs of adjacent angles of the quadrilateral ABCD. (iv) Opposite angles. Two angles of a quadrilateral which are not adjacent angles are known as the opposite angles of the quadrilateral. Thus A, C, and B, D are two pairs of opposite angles of the quadrilateral ABCD. 2. Convex and concave quadrilaterals (i) Convex quadrilateral. A quadrilateral in which the measure of each angle is less than 180º is called a convex quadrilateral. (ii) Concave quadrilateral. A quadrilateral in which the measure of at least one of the angles is more than 180º is called a concave quadrilateral. In the above figure, PQRS is a concave quadrilateral in which S > 180º. In this chapter, by a quadrilateral, we would mean a convex quadrilateral. 3. Interior and exterior of a quadrilateral Consider a quadrilateral ABCD. It divides the whole plane into three parts. (i) The part of the plane lying inside the boundary ABCD is called the interior of the quadrilateral ABCD. Each point of this part is called an interior point of the quadrilateral. In the given figure, the points P, Q, R are the interior points of the quadrilateral ABCD. (ii) The part of the plane lying outside the boundary ABCD is called the exterior of the quadrilateral ABCD. Each point of this part is called an exterior point of the quadrilateral. In the given figure, the points L and M are the exterior points of the quadrilateral ABCD. (iii) The boundary ABCD. Clearly, the point E lies on the quadrilateral ABCD. Quadrilateral region. The interior of the quadrilateral ABCD together with its boundary is called the quadrilateral region ABCD. 4. Angle sum property of a quadrilateral. The sum of the angles of a quadrilateral is 360º. 5. Various Types of Quadrilaterals : 1. Trapezium. A quadrilateral having one and only one pair of parallel sides is called a trapezium. In the adjacent figure, ABCD is a trapezium in which AB || DC. A trapezium is said to be an isosceles trapezium if its nonparallel sides are equal. In the adjoining figure, PQRS is an isosceles trapezium in which PQ || SR and PS = QR. Remark. The diagonals of an isosceles trapezium are always equal. 2. Parallelogram . A quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram. In the given figure, ABCD is a parallelogram in which AB || DC and AD || BC. We denote it by ||gm ABCD. Properties of a parallelogram : (i) The opposite sides of a ||gm are equal and parallel. (ii) The opposite angles of a ||gm are equal. (iii) The diagonals of a ||gm bisect each other. Thus, in a ||gm ABCD, we have (i) AB = DC, AD = BC and AB || DC, AD || BC. (ii) BAD = BCD and ABC = ADC. Page 3 Points to Remember : 1. Quadrilateral. A simple closed figure bounded by four line segments is called a quadrilateral. In the adjacent figure, ABCD is a quadrilateral. A quadrilateral ABCD has : (i) Four vertices, namely A, B, C and D. (ii) Four sides, namely AB, BC, CD and DA. (iii) Four angles, namely DAB, ABC, BCD and CDA, to be denoted by A, B, C and D respectively. (iv) Two diagonals, namely AC and BD. (i) Adjacent sides. Two sides of a quadrilateral which have a common end point are called its adjacent sides. Thus AB, BC ; BC, CD ; CD, DA and DA, AB are four pairs of adjacent sides of the quadrilateral ABCD. (ii) Opposite sides. Two sides of a quadrilateral are called its opposite sides if they do not have a common end point. Thus AB, DC and AD, BC are two pairs of the opposite sides of the quadrilateral ABCD. (iii) Adjacent angles. Two angles of a quadrilateral having one common side are called its adjacent angles. Thus A, B ; B, C ; C, D and D, A are four pairs of adjacent angles of the quadrilateral ABCD. (iv) Opposite angles. Two angles of a quadrilateral which are not adjacent angles are known as the opposite angles of the quadrilateral. Thus A, C, and B, D are two pairs of opposite angles of the quadrilateral ABCD. 2. Convex and concave quadrilaterals (i) Convex quadrilateral. A quadrilateral in which the measure of each angle is less than 180º is called a convex quadrilateral. (ii) Concave quadrilateral. A quadrilateral in which the measure of at least one of the angles is more than 180º is called a concave quadrilateral. In the above figure, PQRS is a concave quadrilateral in which S > 180º. In this chapter, by a quadrilateral, we would mean a convex quadrilateral. 3. Interior and exterior of a quadrilateral Consider a quadrilateral ABCD. It divides the whole plane into three parts. (i) The part of the plane lying inside the boundary ABCD is called the interior of the quadrilateral ABCD. Each point of this part is called an interior point of the quadrilateral. In the given figure, the points P, Q, R are the interior points of the quadrilateral ABCD. (ii) The part of the plane lying outside the boundary ABCD is called the exterior of the quadrilateral ABCD. Each point of this part is called an exterior point of the quadrilateral. In the given figure, the points L and M are the exterior points of the quadrilateral ABCD. (iii) The boundary ABCD. Clearly, the point E lies on the quadrilateral ABCD. Quadrilateral region. The interior of the quadrilateral ABCD together with its boundary is called the quadrilateral region ABCD. 4. Angle sum property of a quadrilateral. The sum of the angles of a quadrilateral is 360º. 5. Various Types of Quadrilaterals : 1. Trapezium. A quadrilateral having one and only one pair of parallel sides is called a trapezium. In the adjacent figure, ABCD is a trapezium in which AB || DC. A trapezium is said to be an isosceles trapezium if its nonparallel sides are equal. In the adjoining figure, PQRS is an isosceles trapezium in which PQ || SR and PS = QR. Remark. The diagonals of an isosceles trapezium are always equal. 2. Parallelogram . A quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram. In the given figure, ABCD is a parallelogram in which AB || DC and AD || BC. We denote it by ||gm ABCD. Properties of a parallelogram : (i) The opposite sides of a ||gm are equal and parallel. (ii) The opposite angles of a ||gm are equal. (iii) The diagonals of a ||gm bisect each other. Thus, in a ||gm ABCD, we have (i) AB = DC, AD = BC and AB || DC, AD || BC. (ii) BAD = BCD and ABC = ADC. (iii) If the diagonals AC and BD intersect at O, then OA = OC and OB = OD. 3. Rhombus. A parallelogram in which all the sides are equal is called a rhombus. In the given figure, ABCD is a rhombus in which AB || DC, AD || BC and AB = BC = CD = DA. Properties of a rhombus : (i) The opposite sides of a rhombus are parallel. (ii) All the sides of a rhombus are equal. (iii) The opposite angles of a rhombus are equal. (iv) The diagonals of a rhombus bisect each other at right angles. Thus, in a rhombus ABCD, we have : (i) AB || DC and AD || BC. (ii) AB = BC = CD = DA. (iii) DAB = BCD and ABC = CDA. (iv) Let the diagonals AC and BD intersect at O. Then, OA = OC, OB = OD and AOB = COD = BOC = AOD = 1 right angle. 4. Rectangle. A parallelogram in which each angle is a right angle, is called a rectangle. In the given figure, ABCD is a rectangle. In which AB || DC, AD || BC and A = B = C = D = 90º. Properties of a rectangle : (i) Opposite sides of a rectangle are equal and parallel. (ii) Each angle of a rectangle is 90º. (iii) Diagonals of a rectangle are equal. Thus, in a rectangle ABCD, we have : (i) AB = DC, AD = BC and AB || DC, AD || BC. (ii) A = B = C = D = 1 right angle. (iii) Diagonal AC = diagonal BD. 5. Square. A parallelogram in which all the sides are equal and each angle is a right angle, is called a square. In the given figure, ABCD is a square in which AB = BC = CD = DA and A = B = C = D = 90º. Properties of a square : (i) The sides of a square are all equal. (ii) Each angle of a square is 90º. (iii) The diagonals of a square are equal and bisect each other at right angle. Thus, in a square ABCD, we have : (i) AB = BC = CD = DA. Page 4 Points to Remember : 1. Quadrilateral. A simple closed figure bounded by four line segments is called a quadrilateral. In the adjacent figure, ABCD is a quadrilateral. A quadrilateral ABCD has : (i) Four vertices, namely A, B, C and D. (ii) Four sides, namely AB, BC, CD and DA. (iii) Four angles, namely DAB, ABC, BCD and CDA, to be denoted by A, B, C and D respectively. (iv) Two diagonals, namely AC and BD. (i) Adjacent sides. Two sides of a quadrilateral which have a common end point are called its adjacent sides. Thus AB, BC ; BC, CD ; CD, DA and DA, AB are four pairs of adjacent sides of the quadrilateral ABCD. (ii) Opposite sides. Two sides of a quadrilateral are called its opposite sides if they do not have a common end point. Thus AB, DC and AD, BC are two pairs of the opposite sides of the quadrilateral ABCD. (iii) Adjacent angles. Two angles of a quadrilateral having one common side are called its adjacent angles. Thus A, B ; B, C ; C, D and D, A are four pairs of adjacent angles of the quadrilateral ABCD. (iv) Opposite angles. Two angles of a quadrilateral which are not adjacent angles are known as the opposite angles of the quadrilateral. Thus A, C, and B, D are two pairs of opposite angles of the quadrilateral ABCD. 2. Convex and concave quadrilaterals (i) Convex quadrilateral. A quadrilateral in which the measure of each angle is less than 180º is called a convex quadrilateral. (ii) Concave quadrilateral. A quadrilateral in which the measure of at least one of the angles is more than 180º is called a concave quadrilateral. In the above figure, PQRS is a concave quadrilateral in which S > 180º. In this chapter, by a quadrilateral, we would mean a convex quadrilateral. 3. Interior and exterior of a quadrilateral Consider a quadrilateral ABCD. It divides the whole plane into three parts. (i) The part of the plane lying inside the boundary ABCD is called the interior of the quadrilateral ABCD. Each point of this part is called an interior point of the quadrilateral. In the given figure, the points P, Q, R are the interior points of the quadrilateral ABCD. (ii) The part of the plane lying outside the boundary ABCD is called the exterior of the quadrilateral ABCD. Each point of this part is called an exterior point of the quadrilateral. In the given figure, the points L and M are the exterior points of the quadrilateral ABCD. (iii) The boundary ABCD. Clearly, the point E lies on the quadrilateral ABCD. Quadrilateral region. The interior of the quadrilateral ABCD together with its boundary is called the quadrilateral region ABCD. 4. Angle sum property of a quadrilateral. The sum of the angles of a quadrilateral is 360º. 5. Various Types of Quadrilaterals : 1. Trapezium. A quadrilateral having one and only one pair of parallel sides is called a trapezium. In the adjacent figure, ABCD is a trapezium in which AB || DC. A trapezium is said to be an isosceles trapezium if its nonparallel sides are equal. In the adjoining figure, PQRS is an isosceles trapezium in which PQ || SR and PS = QR. Remark. The diagonals of an isosceles trapezium are always equal. 2. Parallelogram . A quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram. In the given figure, ABCD is a parallelogram in which AB || DC and AD || BC. We denote it by ||gm ABCD. Properties of a parallelogram : (i) The opposite sides of a ||gm are equal and parallel. (ii) The opposite angles of a ||gm are equal. (iii) The diagonals of a ||gm bisect each other. Thus, in a ||gm ABCD, we have (i) AB = DC, AD = BC and AB || DC, AD || BC. (ii) BAD = BCD and ABC = ADC. (iii) If the diagonals AC and BD intersect at O, then OA = OC and OB = OD. 3. Rhombus. A parallelogram in which all the sides are equal is called a rhombus. In the given figure, ABCD is a rhombus in which AB || DC, AD || BC and AB = BC = CD = DA. Properties of a rhombus : (i) The opposite sides of a rhombus are parallel. (ii) All the sides of a rhombus are equal. (iii) The opposite angles of a rhombus are equal. (iv) The diagonals of a rhombus bisect each other at right angles. Thus, in a rhombus ABCD, we have : (i) AB || DC and AD || BC. (ii) AB = BC = CD = DA. (iii) DAB = BCD and ABC = CDA. (iv) Let the diagonals AC and BD intersect at O. Then, OA = OC, OB = OD and AOB = COD = BOC = AOD = 1 right angle. 4. Rectangle. A parallelogram in which each angle is a right angle, is called a rectangle. In the given figure, ABCD is a rectangle. In which AB || DC, AD || BC and A = B = C = D = 90º. Properties of a rectangle : (i) Opposite sides of a rectangle are equal and parallel. (ii) Each angle of a rectangle is 90º. (iii) Diagonals of a rectangle are equal. Thus, in a rectangle ABCD, we have : (i) AB = DC, AD = BC and AB || DC, AD || BC. (ii) A = B = C = D = 1 right angle. (iii) Diagonal AC = diagonal BD. 5. Square. A parallelogram in which all the sides are equal and each angle is a right angle, is called a square. In the given figure, ABCD is a square in which AB = BC = CD = DA and A = B = C = D = 90º. Properties of a square : (i) The sides of a square are all equal. (ii) Each angle of a square is 90º. (iii) The diagonals of a square are equal and bisect each other at right angle. Thus, in a square ABCD, we have : (i) AB = BC = CD = DA. (ii) A = B = C = D = 90º. (iii) Diagonal AC = diagonal BD. 6. Kite. A quadrilateral which has two pairs of equal adjacent sides but unequal opposite sides, is called a kite. In the given figure ABCD is a kite in which CB = CD and AB = AD but AD BC and AB CD. ( ) EXERCISE 17 A Q.1. In adjacent figure, a quadrilateral has been shown. Name : (i) its diagonal, (ii) two pairs of opposite sides, (iii) two pairs of opposite angles, (iv) two pairs of adjacent sides, (v) two pairs of adjacent angles. D A B C Sol. In the figure, a quadrilateral (i) Its diagonals are AC and BD (ii) Two pairs of opposite sides are AB, CD and AD, BC (iii) Two pairs of opposite angles are A, C and B, D (iv) Two pairs of adjacents sides are AB, BC and CD and DA (v) Two pairs of adjacents angles are A, B and B, C Q. 2. Draw a parallelogram ABCD in which AB = 6·5 cm AD = 4·8 cm and BAD = 70º. Measure its diagonals. Sol. Steps of construction : (i) Draw a line segment AB = 6·5 cm. (ii) At A, draw a ray AE making an angle of 70º with the help of the protractor and cut off AD = 4·8 cm. (iii) With centre B and radius 4·8 cm and with centre D and radius 6·5 cm, draw two arcs intersecting each other at C. (iv) Join BC and DC. Then ABCD is the required parallelogram. (v) Join AC and BD which measures 9·3 cm and 6·6 cm respectively. Q. 3. Two sides of a parallelogram is 4 : 3. If its perimeter is 56 cm, find the lengths of the sides of the parallelogram. Sol. Perimeter of the parallelogram = 56 cm Ratio in sides = 4 : 3 Let first side = 4 x Then second side = 3 x Perimeter = 2 × sum of two sides 56 = 2 × (4 x + 3 x) 7 x × 2 = 56 Page 5 Points to Remember : 1. Quadrilateral. A simple closed figure bounded by four line segments is called a quadrilateral. In the adjacent figure, ABCD is a quadrilateral. A quadrilateral ABCD has : (i) Four vertices, namely A, B, C and D. (ii) Four sides, namely AB, BC, CD and DA. (iii) Four angles, namely DAB, ABC, BCD and CDA, to be denoted by A, B, C and D respectively. (iv) Two diagonals, namely AC and BD. (i) Adjacent sides. Two sides of a quadrilateral which have a common end point are called its adjacent sides. Thus AB, BC ; BC, CD ; CD, DA and DA, AB are four pairs of adjacent sides of the quadrilateral ABCD. (ii) Opposite sides. Two sides of a quadrilateral are called its opposite sides if they do not have a common end point. Thus AB, DC and AD, BC are two pairs of the opposite sides of the quadrilateral ABCD. (iii) Adjacent angles. Two angles of a quadrilateral having one common side are called its adjacent angles. Thus A, B ; B, C ; C, D and D, A are four pairs of adjacent angles of the quadrilateral ABCD. (iv) Opposite angles. Two angles of a quadrilateral which are not adjacent angles are known as the opposite angles of the quadrilateral. Thus A, C, and B, D are two pairs of opposite angles of the quadrilateral ABCD. 2. Convex and concave quadrilaterals (i) Convex quadrilateral. A quadrilateral in which the measure of each angle is less than 180º is called a convex quadrilateral. (ii) Concave quadrilateral. A quadrilateral in which the measure of at least one of the angles is more than 180º is called a concave quadrilateral. In the above figure, PQRS is a concave quadrilateral in which S > 180º. In this chapter, by a quadrilateral, we would mean a convex quadrilateral. 3. Interior and exterior of a quadrilateral Consider a quadrilateral ABCD. It divides the whole plane into three parts. (i) The part of the plane lying inside the boundary ABCD is called the interior of the quadrilateral ABCD. Each point of this part is called an interior point of the quadrilateral. In the given figure, the points P, Q, R are the interior points of the quadrilateral ABCD. (ii) The part of the plane lying outside the boundary ABCD is called the exterior of the quadrilateral ABCD. Each point of this part is called an exterior point of the quadrilateral. In the given figure, the points L and M are the exterior points of the quadrilateral ABCD. (iii) The boundary ABCD. Clearly, the point E lies on the quadrilateral ABCD. Quadrilateral region. The interior of the quadrilateral ABCD together with its boundary is called the quadrilateral region ABCD. 4. Angle sum property of a quadrilateral. The sum of the angles of a quadrilateral is 360º. 5. Various Types of Quadrilaterals : 1. Trapezium. A quadrilateral having one and only one pair of parallel sides is called a trapezium. In the adjacent figure, ABCD is a trapezium in which AB || DC. A trapezium is said to be an isosceles trapezium if its nonparallel sides are equal. In the adjoining figure, PQRS is an isosceles trapezium in which PQ || SR and PS = QR. Remark. The diagonals of an isosceles trapezium are always equal. 2. Parallelogram . A quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram. In the given figure, ABCD is a parallelogram in which AB || DC and AD || BC. We denote it by ||gm ABCD. Properties of a parallelogram : (i) The opposite sides of a ||gm are equal and parallel. (ii) The opposite angles of a ||gm are equal. (iii) The diagonals of a ||gm bisect each other. Thus, in a ||gm ABCD, we have (i) AB = DC, AD = BC and AB || DC, AD || BC. (ii) BAD = BCD and ABC = ADC. (iii) If the diagonals AC and BD intersect at O, then OA = OC and OB = OD. 3. Rhombus. A parallelogram in which all the sides are equal is called a rhombus. In the given figure, ABCD is a rhombus in which AB || DC, AD || BC and AB = BC = CD = DA. Properties of a rhombus : (i) The opposite sides of a rhombus are parallel. (ii) All the sides of a rhombus are equal. (iii) The opposite angles of a rhombus are equal. (iv) The diagonals of a rhombus bisect each other at right angles. Thus, in a rhombus ABCD, we have : (i) AB || DC and AD || BC. (ii) AB = BC = CD = DA. (iii) DAB = BCD and ABC = CDA. (iv) Let the diagonals AC and BD intersect at O. Then, OA = OC, OB = OD and AOB = COD = BOC = AOD = 1 right angle. 4. Rectangle. A parallelogram in which each angle is a right angle, is called a rectangle. In the given figure, ABCD is a rectangle. In which AB || DC, AD || BC and A = B = C = D = 90º. Properties of a rectangle : (i) Opposite sides of a rectangle are equal and parallel. (ii) Each angle of a rectangle is 90º. (iii) Diagonals of a rectangle are equal. Thus, in a rectangle ABCD, we have : (i) AB = DC, AD = BC and AB || DC, AD || BC. (ii) A = B = C = D = 1 right angle. (iii) Diagonal AC = diagonal BD. 5. Square. A parallelogram in which all the sides are equal and each angle is a right angle, is called a square. In the given figure, ABCD is a square in which AB = BC = CD = DA and A = B = C = D = 90º. Properties of a square : (i) The sides of a square are all equal. (ii) Each angle of a square is 90º. (iii) The diagonals of a square are equal and bisect each other at right angle. Thus, in a square ABCD, we have : (i) AB = BC = CD = DA. (ii) A = B = C = D = 90º. (iii) Diagonal AC = diagonal BD. 6. Kite. A quadrilateral which has two pairs of equal adjacent sides but unequal opposite sides, is called a kite. In the given figure ABCD is a kite in which CB = CD and AB = AD but AD BC and AB CD. ( ) EXERCISE 17 A Q.1. In adjacent figure, a quadrilateral has been shown. Name : (i) its diagonal, (ii) two pairs of opposite sides, (iii) two pairs of opposite angles, (iv) two pairs of adjacent sides, (v) two pairs of adjacent angles. D A B C Sol. In the figure, a quadrilateral (i) Its diagonals are AC and BD (ii) Two pairs of opposite sides are AB, CD and AD, BC (iii) Two pairs of opposite angles are A, C and B, D (iv) Two pairs of adjacents sides are AB, BC and CD and DA (v) Two pairs of adjacents angles are A, B and B, C Q. 2. Draw a parallelogram ABCD in which AB = 6·5 cm AD = 4·8 cm and BAD = 70º. Measure its diagonals. Sol. Steps of construction : (i) Draw a line segment AB = 6·5 cm. (ii) At A, draw a ray AE making an angle of 70º with the help of the protractor and cut off AD = 4·8 cm. (iii) With centre B and radius 4·8 cm and with centre D and radius 6·5 cm, draw two arcs intersecting each other at C. (iv) Join BC and DC. Then ABCD is the required parallelogram. (v) Join AC and BD which measures 9·3 cm and 6·6 cm respectively. Q. 3. Two sides of a parallelogram is 4 : 3. If its perimeter is 56 cm, find the lengths of the sides of the parallelogram. Sol. Perimeter of the parallelogram = 56 cm Ratio in sides = 4 : 3 Let first side = 4 x Then second side = 3 x Perimeter = 2 × sum of two sides 56 = 2 × (4 x + 3 x) 7 x × 2 = 56 14 x = 56 x 56 14 4 First side = 4 x = 4 × 4 = 16 cm and second side = 3 x = 3 × 4 = 12 cm. Ans. Q. 4. Name each of the following parallelograms : (a) The diagonals are equal and the adjacent sides are unequal. (b) The diagonals are equal and the adjacent sides are equal. (c) The diagonals are unequal and the adjacent sides are equal. Sol. (a) . . . A parallelograms whose diagonals are equal and adjacent sides are unequal, is a rectangle. (b) . . . A parallelogram whose diagonal are equal and also side are equal, is a square. (c) . . . A parallelogram whose diagonal are unequal but adjacent sides are equal is a rhombus. Q. 5. What is a trapezium ? When do you call a trapezium an isosceles trapezium ? Draw an isosceles trapezium. Measure its sides and angles. Sol. A quadrilateral whose one pair of opposite sides are equal but other pair non parallel, is called a trapezium. When the non-parallel sides of a trapezium are equal, then it is called an isosceles trapezium. ABCD is an isosceles trapezium in which AD = BC Then DAB = CBA On measuring, AD = BC = 3 cm and DAB = CBA = 60º D C B A Q. 6. Which of the following statements are true and which are false ? (a) The diagonals of a parallelogram are equal. (b) The diagonals of a rectangle are perpendicular to each other. (c) The diagonals of a rhombus are equal. Sol. (a) False . . . Diagonals of a parallelogram are not equal. (b) False . . . Diagonals of a rectangle do not bisect each other at right angles. (c) False . . . Diagonals of a rhombus are not equal. Q.7. Given reasons for the following : (a) A square can be thought of as a special rectangle. (b) A square can be thought of as a special rhombus. (c) A rectangle can be thought of as a special parallelogram. (d) Square is also a parallelogram. Sol. (a) Because if each side of a rectangle are equal it is called a square. (b) Square is a special rhombus if its each angle is equal i.e., of 90º. (c) If in a parallelogram, if each angle is of 90º, it is called a rectangle. (d) A square is a parallelogram whose each side and each angle are equal. Q.8. A figure is said to be regular if its sides are equal in length and angles are equal in measure. What do you mean by a regular quadrilateral ? Sol. A regular quadrilateral is a quadrilateral if its each side and angles are equal square is a regular quadrilateral. ( ) EXERCISE 17 B Objective questions Mark ( ) against the correct answer in each of the following : Q. 1. The sum of all the angles of a quadrilateral is (a) 180º (b) 270º (c) 360º (d) 400º Sol. (c) . . . Sum of angles of a quadrilateral is 360º. Q.2. The three angles of a quadrilateral are 80º, 70º and 120º. The fourth angle is (a) 110º (b) 100º (c) 90º (d) 80º Sol. Sum of 4 angles of a quadrilateral = 360ºRead More

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