Understanding Quadrilaterals Parallelogram: A quadrilateral with each pair of opposite sides parallel. Properties: (1) Opposite sides are equal. (2) Opposite angles are equal. (3) Diagonals bisect one another. Rhombus: A parallelogram with sides of equal length. Properties: (1) All the properties of a parallelogram. (2) Diagonals are perpendicular to each other.
Rectangle: A parallelogram with a right angle. Properties: (1) All the properties of a parallelogram. (2) Each of the angles is a right angle. (3) Diagonals are equal. Square: A rectangle with sides of equal length. Properties: All properties of rectangle. Kite: A quadrilateral with exactly two pairs of equal consecutive sides Properties: (1) The diagonals are perpendicular to one another (2) One of the diagonals bisects the other. (3) In the figure B = D but A C
Diagonal: A diagonal is a line connecting two no-consecutive vertices of a polygon. Angle Sum of a Polygon: The angle sum of a polygon is given by following formula: (n-2)180, where n is the number of sides of a polygon. Sum of exterior angles of a polygon: This is always 360, no matter what is the number of sides in a given polygon. Take example of a triangle. All vertex of a triangle will make to exterior angles. As you know the angle sum of a triangle is 180, so three exterior angles will sum up to 180 and two sets of three exterior angles each will sum up to 360. In a rectangle all exterior angles are of 90, hence their sum is equal to 360. EXERCISE 1 1. How many diagonals does each of the following have? (a) A convex quadrilateral (b) A regular hexagon (c) A triangle Answer: (a) 2 (b) 9 (c) 0 2. What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? Answer: Using the formula: (n-2)180 = (4-2)180 = 360. For concave quadrilateral also angle sum will be same because number of sides is same. 3. What can you say about the angle sum of a convex polygon with number of sides? (a) 7 (b) 8 (c) 10 Answer: (a) Number of sides = 7 So, Angle Sum= (n 2) 180 = (7 2) 180 = 900 (b) Number of sides = 8 So, Angle Sum= (8 2) 180 = 1080 (c ) Number of sides = 10 So, Angle Sum = (10 2) 180 = 1440 4. What is a regular polygon? State the name of a regular polygon of (i) 3 sides (ii) 4 sides (iii) 6 sides
Answer: In a regular polygon all sides are of equal length. (i) Triangle (ii) Quadrilateral (iii) Hexagon 5. Find the angle measure x in the following figures. (a) Answer: As we know that angle sum of a quadrilateral is 360. So, 50 +130 +120 +x = 360 Or, x = 360-300 = 60 (b)
Answer: Angles on one side of a line always add up to 180, so third angle in the given quadrilateral is 90 Now, x+70 +60 +90 = 360 Or, x = 360-220 = 140 (c) Answer: Angle adjacent to 70 = 180-70 = 110 Angle adjacent to 60 = 180-60 =120 Now, 2x+110 +120 +30 = 360 Or, 2x = 360-260 = 100 Or, x = 50 (d) Answer: Angle sum of pentagon (n-2)180 = (5-2)180 =3 X 180 = 540 Hence, measurement of 1 angle of a pentagon = 540 5 = 108 (e) x + y + z =?
Answer: As we know sum of all external angle of a polygon is always 360. So, x + y + z = 360 Alternate Method: The angle adjacent to z = 180 -(90 +30 ) = 60 Now, x = 180-90 = 90 y = 180-30 = 150 z = 180-60 = 120 Now, x + y + z = 90 + 150 + 120 = 360 (f) w + x + y + z =?
Answer: Sum of all the external angles of a polygon is always 360. Hence, w + x + y + z = 360 EXERCISE 2 1. Find x in the following figures. (a) Answer: x = 360 (125 + 125 ) = 110 Because sum of all exterior angles of a polygon is always 360 (b)
Answer: x = 360 (70 + 60 + 90 + 90 ) = 50 Another exterior angle is 90 because angles on the same side of a line always add up to 180 2. Find the measure of each exterior angle of a regular polygon of (i) 9 sides (ii) 15 sides Answer: (i) Sum of all exterior angles of a polygon = 360 So, 1 exterior angle of a polygon of 9 sides = 360 9 = 40 (ii) 1 exterior angle of a polygon of 15 sides = 360 15 = 24 3. How many sides does a regular polygon have if the measure of an exterior angle is 24? 360 360 Answer: Number of sides of a polygon = = = 15 exteriorangle 24 4. How many sides does a regular polygon have if each of its interior angles is 165? Answer= Exterior Angle = 180 -interior angle = 180 165 = 15 360 Number of Sides = = 24 15 5. (a) Is it possible to have a regular polygon with measure of each exterior angle as 22?
360 360 3 Answer: Number of Sides = = = 17 exteriorangle 22 11 As the final answer is not a whole number so there is no possibility of a polygon with 1 exterior angle measuring 22. (b) Can it be an interior angle of a regular polygon? Why? Answer: If interior angle is 22 then the exterior angle = 180-22 =158 On dividing 360 by 158 we can t get answer in whole number, so such a polygon is not possible. 6. (a) What is the minimum interior angle possible for a regular polygon? Why? (b) What is the maximum exterior angle possible for a regular polygon? Answer: The polygon with minimum number of sides is a triangle, and each angle of an equilateral triangle measures 60, so 60 is the minimum value of the possible interior angle for a regular polygon. For an equilateral triangle the exterior angle is 180-60 =120 and this is the maximum possible value of an exterior angle for a regular polygon. EXERCISE 3 1. Given a parallelogram ABCD. Complete each statement along with the definition or property used. (i) AD = Opposite Sides are Equal (ii) DCB = Opposite Angles are equal (iii) OC = Diagonals Bisect Each Other (iv) m DAB + m CDA = 180
2. Consider the following parallelograms. Find the values of the unknowns x, y, z. (i) Answer: x = 180 100 = 80 As Opposite angles are equal in a parallelogram So, y = 100 And, z = 80 (ii) Answer: x, y and z will be complementary to 50. So, Required angle=180-50 =130
(iii) Answer: z being opposite angle= 80 x and y are complementary, x and y =180-80 =100 (iv) Answer: As angles on one side of a line are always complementary So, x=90 y = 180 -(90 +30 )=60 The top vertex angle of the above figure = 60 2=120 Hence, bottom vertex Angle = 120 and z=60 (v) Answer: y= 112, as opposite angles are equal in a parallelogram x= 180 -(112-40 )=28
As adjacent angles are complementary so angle of the bottom left vertex =180-112 =68 So, z=68-40 =28 Another way of solving this is as follows: As angles x and z are alternate angles of a transversal so they are equal in measurement. 3. Can a quadrilateral ABCD be a parallelogram if (i) D + B = 180? (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) A = 70 and C = 65? Answer: (i)it can be, but not always as you need to look for other criteria as well. (ii) In a parallelogram opposite sides are always equal, here AD BC, so its not a parallelogram. (iii) Here opposite angles are not equal, so it is not a parallelogram. 5. The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram. Answer: Opposite angles of a parallelogram are always add upto 180. So, 180 = 3x + 2x 5x = 180 x = 36 So angles are; 36 3 = 108 and 36 2 = 72 6. Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram. Answer: 90, as they add up to 180 7. The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them. Answer: Angle opposite to y = 180-70 =110 Hence, y = 110 x=180 -(110 +40 )=30, (triangle s angle sum) z=30 (Alternate angle of a transversal)
8. The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm) Answer: As opposite sides are equal in a parallelogram SO, 3 y 1 = 26 3 y = 27 y = 9 Similarly, 3 x = 18 x = 6 Answer: As you know diagonals bisect each other in a parallelogram. So, y + 7 = 20 y = 20 7 = 13 Now, x + y = 16 x + 13 = 16 x = 16 13 = 3 9. In the given figure both RISK and CLUE are parallelograms. Find the value of x.
Answer: In parallelogram RISK ISK = 180 120 = 60 Similarly, in parallelogram CLUE CEU = 180 70 = 110 Now, in the triangle x = 180 (110 60 ) = 10 EXERCISE 4 1. State whether True or False. (a) All rectangles are squares All squares are rectangles but all rectangles can t be squares, so this statement is false. (b) All kites are rhombuses. All rhombuses are kites but all kites can t be rhombus. (c) All rhombuses are parallelograms True. (d) All rhombuses are kites. True (e) All squares are rhombuses and also rectangles True; squares fulfill all criteria of being a rectangle because all angles are right angle and opposite sides are equal. Similarly, they fulfill all criteria of a rhombus, as all sides are equal and their diagonals bisect each other. (f) All parallelograms are trapeziums. False; All trapeziums are parallelograms, but all parallelograms can t be trapezoid.
(g) All squares are not parallelograms. False; all squares are parallelograms (h) All squares are trapeziums. True; 2. Identify all the quadrilaterals that have. (a) four sides of equal length (b) four right angles Answer: (a) If all four sides are equal then it can be either a square or a rhombus. (b) All four right angles make it either a rectangle or a square. 3. Explain how a square is. (i) a quadrilateral (ii) a parallelogram (iii) a rhombus (iv) a rectangle Answer: (i) Having four sides makes it a quadrilateral (ii) Opposite sides are parallel so it is a parallelogram (iii) Diagonals bisect each other so it is a rhombus (iv) Opposite sides are equal and angles are right angles so it is a rectangle. 4. Name the quadrilaterals whose diagonals. (i) bisect each other (ii) are perpendicular bisectors of each other (iii) are equal Answer: Rhombus; because, in a square or rectangle diagonals don t intersect at right angles. 5. Explain why a rectangle is a convex quadrilateral. Answer: Both diagonals lie in its interior, so it is a convex quadrilateral. 6. ABC is a right-angled triangle and O is the mid point of the side opposite to the right angle. Explain why O is equidistant from A, B and C.
Answer: If we extend BO to D, we get a rectangle ABCD. Now AC and BD are diagonals of the rectangle. In a rectangle diagonals are equal and bisect each other. So, AC= BD AO= OC BO= OD And AO=OC=BO=OD So, it is clear that O is equidistant from A, B and C.