202110722-PERFORM-STUDENT-WORKBOOK-MATHEMATICS-G09-FY_Optimized

PRACTICE SHEET - 23 (PS-23) 1. Draw the graph 4x – 3y + 1 = 13. 2. Draw the geometrical representation of the following equations in the same graph: i) x + y = 2 ii) x + y = –2 3. Draw a graph of 5x – 3y – 15 = 0. Check whether the graph passes through the origin. 4. Draw the graph 4x – 3y =0. From the graph, check whether the following points are present on the line: i) (-2,0)   ii) (5,1) 5. A graph is available and its equation is not clearly known. The possible equations are given below. State the equation which is represented by the graph. i)3x+2y=-5 ii) 3x-2y=5 6. In a festive season, a shop keeper wants to pack two types of fruits in a box and the total cost should always remain Rs. 200 and final weight is not important. The cost one type of fruit is Rs. 50 per kg and the cost of the second variety is Rs. 100 per kg. Which quadrants of the graph are needed? Draw a graph which would help the shop keeper in finding the amount of fruits he should pack in the box. 7. Geometrically represent the following equations on a number line and in a graph. i) x=4 ii) x-5 8. Draw the graphs of x – y = 0, x = 5 and y = 5. 9. Draw the graph of the following equation and check if it passes through first quadrant: 4 y = 20 − 5x Mention the quadrant in which the line will be present. 37

PRACTICE SHEET - 4 (PS-4) I. Choose the correct option. 1. Which would be the graph of y = x? (A) (B) (C) (D) 2. Abscissa of all points on the y-axis is: (A) 0 (B) 1 (C) -1 (D) None of these 3. The geometric representation of x = -2 meets the x-axis at: (A) (2, 0) (B) (-2, 0) (C) (0, 2) (D) (0, -2) 4. Age of a mother is 5 years more than 7 times the present age of his son. The above statement can be expressed in a linear equation as: (A) x – 7y – 5 = 0 (B) x + 7y – 5 = 0 (C) x + 7y + 5 = 0 (D) x – 7y + 5 = 0 5. The number of solutions, the equation 7x + 3y + 21 = 0 can have is: (A) one only (B) exactly two (C) has no solution (D) infinite 6. If (2, m) lies on the graph of 4x + y - 9 = 0, then m = ? (A) 5 (B) 0 (C) 1 (D) -1 7. In the given graph, OABC is a square whose equations of sides are: (A) x = 4, y = 4, x = -4, y = -4 (B) x = 4, y = 4, x = 0, y = 0 (C) x = -4, y = 4, x = -4, y = 0 (D) x = 4, y = 4, x = -4, y = 0 8. If (20, -a) lies on l whose graph is given then the value of a is: (A) -5 (B) 5 (C) -10 (D) 10 (D) x – y = 2 9. Which of the following equation has graph parallel to y-axis? (A) x = -1 (B) y = -2 (C) x = 1 38

PRACTICE SHEET - 4 (PS-4) 10. Which of the following equation has graph shown in figure? (A) x + y = 0 (B) y = 2x (C) y = 2x + 4 (D) y = x - 4 II. Short answer questions. 1. Write the linear equations in two variables in the form ax + by + c = 0 by using the given values of a, b, c. (i) a = 2, b = -5, c = 1 (ii) a = 5 , b = - 5 , c = 7 7 39 2. The cost of petrol in a city is ` 80 per litre. Write an equation with y representing the number of litres and x representing the total cost (in rupees). Also, draw its graph. 3. Express y in terms of x, given that 5y – 9x = 25. Check whether (2, -6) is a solution of the line. III. Long answer questions. 1. Draw the graph of the equation 2x + y = 5. 2. A cycle shop owner charge ` 200 rent per cycle for one day and ` 50 per day thereafter. If soni had taken a cycle for x days and y be the total amount needs to be paid, write the linear equation in two variables for this situation. Plot its graph and find the amount to be paid. 39

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. Express the given equation in the form 4. Draw the graph of 3x + 2y = 12. (4 Marks) ax + by + c = 0 and determine the values of a, b and c. Mention the variables in the equation. i) x + y –10 = 2x – 3y +14 ii) 2x + 2 y − 22 = x + y − 52 (2 Marks) 2. One box of sweets weighs 1 kg and 1 box of biscuits weighs 0.5 kg. Ravi purchases 10 kgs of sweets & biscuits and finds that the total weight of all boxes is 10 kg. What will be the relation between the number of boxes of sweets and biscuits purchased by Ravi? (2 Marks) 5. Which among the following equations does the given graph represent?  (3 Marks) i) 3x = 6y – 6 ii) x – 2y = –2 iii) 4y = 2x + 4 3. Find four solutions such that none of the variables have zero value. 5x – 4y = 2  (4 Marks) 40

5. Introduction to Euclids Geometry Learning Outcome By the end of this chapter, a student will be able to: • Explain Euclid’s axioms • Explain the Euclid’s Postulates. Concept Map Euclid’s Axioms Euclid’s Postulates Concept of Point, Line, Ray, Line segment, Angle, Application to Problems Plane (Surface) and Solid Polygons, Circles and Geometrical Construction Key Points o Polygon: A simple closed curve made up of only line segments is called a polygon. A regular • A solid has shape, size, and position and can be polygon is both equiangular and equilateral. moved from one place to another. Its boundaries are called surfaces. The surfaces separate one part o  Diagonal: Diagonal is a line segment connecting of the space from another and are said to have two non-consecutive vertices of a polygon. no thickness. The boundaries of the surfaces are curves or straight lines. These lines end in points. o Parallelogram is a quadrilateral whose opposite sides are equal and parallel. • A solid has three dimensions, surface has two di- mensions, a line has one dimension and a point – Rhombus is a quadrilateral with all sides of has zero dimensions. equal length and diagonals are perpendicular bisectors of one another. • Some definitions: o A line segment has two end points and if we – Rectangle is a parallelogram with a right angle. extend the two end points in either direction endlessly we get a line. – Square is a rectangle with sides of equal o Angle: An angle is formed when lines or line length. segments meet. o Ray: Ray is a part of line which has only one end • Euclid’s Statements: point and extends indefinitely in one direction. o  A point is that which has no part. A ray has no definite length. o  A line is breadthless length. o If the perpendicular distance between two lines o  The ends of a line are points. remains constant then such lines are called o A straight line is a line which lies evenly with the parallel lines. o If the angle between two intersecting lines is points on itself. 90⁰, such lines are called perpendicular lines. o  A surface is that which has length and breadth only. o  The edges of a surface are lines. o A plane surface is a surface which lies evenly with the straight lines on itself. • In geometry, we take a point, a line, a plane (plane surface) as undefined terms but are represented as physical models. • Postulates are assumptions that are specific to ge- ometry. • Common notions (Axioms) are assumptions used throughout mathematics and not specifically linked to geometry. 41

5. Introduction to Euclids Geometry • Euclid’s Axioms: • Example: Areas of two geometric shapes can be Things which are compared. equal to the same thing are equal to  Comparison of area of one shape and volume of one another. another is not possible. Example: • Euclid’s Postulates: i) A = P, B = P, means A = B Postulate 1: ii) AB = a CD = a, A straight line may be drawn from any one point to hence AB = CD any other point. If equals are added Example: to equals, the wholes are equal. Example: Axiom 5.1: Example: i) AB = a, BC = b Given two distinct If equals are PQ = a, RS = b points, there is subtracted from then PQ + RS = AB + BC a unique line equals, the Example: passing through remainders are AC = l, AB = a, AC – AB = BS them. equal. = l -a Things which PS = l, QS = a, PS – QS = l - a Postulate 2: coincide with one Then AC – AB = PS – QS A terminated line can be produced indefinitely. another are equal to Example: one another. Example AB = PQ Postulate 3: The whole is greater A circle can be drawn with any centre and any than part. radius. Example: Example ii))100 > 100 2 ii) AC > AB, AC > BC o Things which are Example double of the same AB = CD means 2AB = 2CD things are equal to one another. o Things which are halves of the same tohniengasnaorteheerq.u=al to EAxBamCp=Dle, means AB CD 22 • Magnitudes of same kind can be compared but not of different kind. 42

5. Introduction to Euclids Geometry Postulate 4: Euclidean geometry. It is called spherical geometry. All right angles are equal to one another. In spherical geometry, lines are not straight but Example: are part of great circles. (Great circles are circles obtained by the intersection of sphere and planes passing through the centre of the sphere.) • Euclidean geometry is valid only for the figures in the plane and on curved surfaces it fails. Postulate 5: If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles. Example: PQ intersects AB and CD. Sum of angles 1 , 2 is less than 180, hence the lines intersect on the side of points A, B. • A system of axioms is called consistent, if it is impossible to deduce from these axioms a statement that contradicts any axiom or previously proved statement. Hence, when any system of axioms is given (or stated), it must be ensured that they are consistent. • Statements that are proved using axioms and postulates are called Propositions or Theorems. • Theorem 5.1: Two distinct lines cannot have more than one point in common. • Playfair’s Axiom (A version of Euclid’s Fifthe Postulate) For every line l and for every point P not lying on l, there exists a unique line m passing through P and parallel to l. (or) Two distinct intersecting lines cannot be parallel to the same line. • The geometry of the universe we live in has is a non- 43

5. Introduction to Euclids Geometry Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Pre-requisites • Understanding the meaning of Point, PS – 1 Line, Surface, Plane and Solid. Euclid’s Geometry • Euclid’s Postulates PS – 2 • Euclid’s Axioms Worksheet for \"Introduction to Euclids Geometry\" PS -3 Evaluation with Self Check or Peer ---- Self Evaluation Sheet Check* 44

PRACTICE SHEET - 1 (PS-1) 1. Name the geometry which can be represented by the following figures. 2. Which of the following figures represents a Ray? 3. How many points exist on a line segment? 45

PRACTICE SHEET - 2 (PS-2) 1. How many lines can be drawn through: i) One Point ii) Two points 2. Five points A, B, C, D and E are chosen on a straight line as shown in the figure such that C is the midpoint of AE and BD. Prove that AB = DE. 3. Four points A, B, C and D are chosen in order. Prove that a line passing A, B and D is equal to a line passing through A, C and D. 4. Ravi is asked to draw a line passing through only one point in his book He finds two points on his book and draws two lines each passing through one point. Can Ravi draw parallel lines through the two points? State the axioms/postulates used to find the solution. 5. Saritha and Saroja are asked to draw straight lines on the board. If the lines are not parallel then how many times the lines cross one another. 6. Two lines are drawn perpendicular to a given line AB as shown. State the nature of the lines. 7. How can a circle and square be compared? How can two circles be compared? 46

PRACTICE SHEET - 3 (PS-3) I. Choose the correct option. 1. Which of the following needs a proof: (A) Postulates (B) Proposition (C) Definition (D) Axiom 2. A solid has: (A) no dimension (B) three-dimension (C) two-dimensions (D) one-dimensions 3. Euclid’s axiom 5 is: (A) The things which coincide with one another are equal. (B) If equals are subtracted from equals, the remainders are equal. (C) The whole is greater than the part. (D) None of these. 4. Which of the following is not an undefined term? (A) angle (B) plane (C) point (D) line 5. Two distinct intersecting lines l and m cannot have: (A) one point in common (B) two points in common (C) any point in common (D) None of these 6. For every line n and for every point M (not on n), there does not exist a unique line through M: (A) Which is perpendicular to n (B) which is not parallel to n (C) which is coincident with n (D) None of these 7. If a point C lies between two point A and B such that AC = BC, then: (A) AC = AB (B) AC = 1 AB (C) AB = 1 AC (D) AB = 1 AC 2 3 2 8. In the given figure AC = BD then which of the following axioms shows that AB = CD? (A) Things which are halves of the same things are equal to one another. (B) If equals are subtracted from equals, the remainders are equal. (C) Things which are equal to the same thing are equal to one another. (D) If equals are added to equals, the wholes are equal. 9. In the given figure, if ⇒ 1 + ⇒ 2 < 180° then m and n will eventually meet at: (A) left side of AB (B) right side (C) either side of AB (D) will never meet 10. If AB = CD, CD = EF and EF = PQ then which one of the following is not true? (A) AB = PQ (B) CD = PQ (C) AB = EF (D) AB ≠ CD 47

PRACTICE SHEET - 3 (PS-3) II. Short answer questions. 1. A line cannot be compared with a rectangle. Explain. 2. Three points lie on a line and B lies between A and C, as shown below, then prove that AC– AB = BC. 3. In the given figure B and C are mid-points of AC and BD respectively. If BC is subtracted from AC and BD, then what is the relation between AB and CD? III. Long answer questions. 1. Answer the following questions in integer form. (a) The whole is what times its half? (b) How many dimensions a surface has? (c) A ray has ‘n’ end points? What is the value of ‘n’? (d) How many postulates Euclid had given? 2. (a) In the given figure, QM = 1 PQ, QN = 1 QR and PQ = QR. Show that QM = QN. 22 (b) In the given figure, AC = XD, and C and D are the mid–points of AB and XY respectively. Show that AB = XY. 48

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. Among the given pair of values, Explain which 3. A quadrilateral ABCD is as shown in the figure. one is bigger. If opposite sides are extended in pairs how many triangles will be formed on the side of i) 20 m, 200 ii) 15m2, 15cm iii) 200, Right angle the quadrilateral? Explain. (4 Marks) iv) Diameter and a radius of a given circle. (2 Marks) 2. Define equilateral triangle. Explain all the terms needed to define it. (1 Mark) 4. Can principles of Euclidean geometry be applied to cube, sphere, square, ellipse? (2 Marks) 49

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 5. A line segment AB has midpoint at M. Can two circles be drawn with centers at A, B and passing through M. Whether the diameter of the two circles is equal? (3 Marks) 6. What is the thickness of a line? (1 Mark) 7. Which among Euclid’s axioms is the universal truth?  (1 Mark) 8. Is it always possible to draw a line through any given set of three points? (1 Mark) 50

6. Lines and Angles Learning Outcome By the end of this lesson, a student will be able to: • Calculate the angles formed by lines and the • Explain the different types of angles. transversal. • Calculate angles between the intersecting lines and • Determine the unknown angles and their find relations between them. relationships in triangles. Concept Map Angles Acute Obtuse Reflex Straight Angle Angle Angle Angle Supplementary Complementary Linear Lines angles angles Pair Intersecting Lines Non-intersecting (Parallel) lines Vertically Corresponding Alternate Transversal opposite angles angles angles Theorems & Axioms Adjacent angles and Vertically opposite angles in intersecting lines. Alternate angles, Corresponding angles and Supplementary angles formed by a transversal and a pair of parallel lines. Angles in a triangle Angles in geometrical figures Key Points Acute angle (∠BAC)  Right angle (BAC)  • A portion of a line with two end points is called a Obtuse angle (∠BAC) 0°< ∠BAC<90⁰ line segment. Line segment with end points A, B is   ∠BAC= 90⁰  90⁰ < ∠BAC <180⁰ denoted as AB and its length is denoted as AB. o  A straight angle is equal to 180⁰. o An angle greater than 180⁰ but less than 360⁰ is • A part of a line with one end point is called a Ray. A ray AB is denoted as AB and a line AB is denoted called Reflex angle. as AB . • If three or more points lie on the same line, they are called collinear points otherwise they are called non-collinear points. • An angle is formed when two rays originate from the same end point. The rays making an angle are called arms of the angle and the end point is called vertex of the angle. o  An acute angle measures between 0⁰ and 90⁰. o  Right angle is exactly equal to 90⁰. o An angle greater than 90⁰ but less than 180⁰ is called obtuse angle. 51

6. Lines and Angles Straight angle (∠BAC)      Reflex angle (∠BAC) ∠BAC= 180⁰       • Intersecting lines have a common point. 180⁰ < ∠BAC< 360⁰ • Two lines are perpendicular if they form a right o If the sum of two angles is 90⁰ then they care angle at their point of intersection. called complementary angles. • Two lines are said to be parallel if they do not o If the sum of two angles is 180⁰ then they are intersect each other on extending both the called supplementary angles. lines infinitely. (OR) Two lines are parallel if the • Two angles are called adjacent angles if they have perpendicular distances between them remains same throughout. a common vertex, a common arm and their non • Theorems and Axioms: common arms are on different sides of the common Axiom 6.1 arm. If a ray stands on a When two angles are adjacent, their sum is always line, then the sum of equal to the angle formed by the non adjacent arms. the adjacent angles (OR) If two adjacent angles are supplementary then so formed is 180⁰. they form a linear pair. ∠AOB, ∠BOC are adjacent angles. Axiom 6.2 OC is ray, AB is line, O is ∠BOC, ∠DOC are adjacent angles. If the sum of two point on the line. ∠AOB, ∠COD are non-adjacent angles adjacent angles is ∠AOC + ∠BOC = 180⁰ 180⁰, then the non- ∠AOC + ∠COB = 180⁰, common arms of the AO, OB are non-common From the above figure, angles form a line. arms of adjacent angles and ∠AOC = ∠AOB + BOC, Theorem 6.1 form a straight line. ∠BOD = ∠BOC + ∠DOC. If two lines intersect AB, CD intersects at O. • A linear pair of angles is formed when two lines each other, then the Vertically opposite angles vertically opposite are equal intersect. Two angles are said to form a linear angles are equal. ⇒ ∠AOC = ∠BOD pair if they are adjacent angles formed by two   ∠ AOD = ∠COB intersecting lines. In the figure, ∠CAD and ∠DAB form a linear pair. • Vertically opposite angles are formed when two lines intersect each other and there will be two pairs of vertically opposite angles. ∠AOC and ∠BOD are vertically opposite angles. ∠COB and ∠AOD are vertically opposite angles. • A line which intersects two or more lines at distinct points is called a transversal. Line PQ is the transversal intersecting AB, CD at points R, S forming eight angles as shown. 52

6. Lines and Angles Theorem 6.2 If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal. Theorem 6.3 l is transversal, If a transversal intersects m, n are parallel lines two lines such that a ∠3 = ∠5 ∠4 = ∠6 pair of alternate interior angles is equal, then the two lines are parallel. Theorem 6.4 The angles ∠1, ∠2, ∠7, ∠8 are called exterior angles If a transversal intersects two parallel lines, then and ∠3, ∠4, ∠5, ∠6 are called interior angles. each pair of interior o  Corresponding angles are: angles on the same side ∠1 and ∠5 ∠2 and ∠6 of the transversal is ∠3 and ∠7 ∠4 and ∠8 supplementary. o  Alternate interior angles or alternate angles. ∠4 and ∠6 ∠3 and ∠5 Theorem 6.5 l is transversal, o  Alternate exterior angles If a transversal intersects m, n are parallel lines ∠1 and ∠7  ∠2 and ∠8 two lines such that a ∠4 + ∠5 = 180⁰ o Interior angles on same side of transversal are pair of interior angles ∠3 + ∠6 = 180⁰ on the same side of also called consecutive interior angles or allied the transversal is angled or co-interior angles supplementary, then the ∠4 and ∠5  ∠3 and ∠6 two lines are parallel. Theorem 6.6 • Theorems and Axioms Lines which are parallel to the same line are Axiom 6.3 parallel to each other. If a transversal intersects two parallel lines, then each pair of corresponding angles is equal. This axiom is also called corresponding angles axiom. Axiom 6.4 l is transversal, Line q is parallel to line If a transversal intersects m, n are parallel lines p and line r is parallel to two lines such that a pair ∠1 = ∠5 ∠2 = ∠6 ∠3 = line p. of corresponding angles ∠7 ∠4 = ∠8 Line q and r are parallel is equal, then the two to each other. lines are parallel. 53

6. Lines and Angles • Angle Sum Property Theorem 6.7 The sum of angles of a triangle is 180⁰ ∠ABC + ∠BCA + ∠BAC = 180⁰ Theorem 6.8 If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles. ∠CBA + ∠BAC = ∠ACD Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Pre-requisites • Different types of Angles. PS – 1 Lines and Angles • Intersecting and Non-intersecting lines PS – 2 • Pair of Angles • Parallel lines and a Transversal PS - 3 • Lines parallel to the same line • Angle Sum Property of Triangles PS – 4 Worksheet for “Lines and Angles” PS - 5 Evaluation with Self Check or ---- Self Evaluation Sheet Peer Check* 54

PRACTICE SHEET - 1 (PS-1) 1. Name the type of angle with reason. i) 30⁰ ii) 90.5⁰ iii) 180⁰ iv) 210⁰ 2. In a pair of complementary angles, if one angle is 32⁰, what will be the other angle? 3. Two lines AB and CD are intersecting at O. Identify the following angles from the figure: i) Acute angle       ii) Obtuse angle iii) Straight angle       iv) Adjacent angles v) Non-adjacent angles    vi) Linear pair vii) Vertically opposite angle viii) S upplementary angles. 4. In the given figure, if AB and CD are parallel lines, and AC= a then what is the value of BD? 5. In the given figure, which lines are transversals? 6. State the corresponding angles and interior angles in the figure below. Identify the transversal. 55

PRACTICE SHEET - 2 (PS-2) 1. If two lines intersect to create 4 equal angles, then what is the angle between them? Are the lines perpendicular? 2. Two lines AB and CD are intersecting at point O. If the ∠AOB is 44⁰ find the value of ∠AOC, ∠COD, ∠BOD: i) Without considering vertically opposite angles. ii) Using principle of linear pairs and considering vertically opposite angles. 3. “If two lines intersect, then the ratio of the four angles formed is 1:2:3:4.” Check if this statement is true. 4. Two lines are intersecting such that the ratio of the larger and smaller angle is 3:1. Find the angles. 5. On a straight line CD, two lines are drawn through points A, B such that the OA is the bisector of ∠BOD, find all the angles formed. 6. Four rays start from O as shown. Check if any two rays can combine to form a straight line. 7. Three rays OA, OB and OC are starting from a point O as shown in the figure. If AD is a straight line and ∠COD = 90⁰ , find the values of a, b and c. 8. A pair of lines BF and DH intersects at O. Through O, a pair of perpendicular lines AE, CG are drawn such that AE bisects the angle ∠BOH and ∠BOA = a. Find the values of the angles b, c, d, e, f and h in terms of a. 56

PRACTICE SHEET - 3 (PS-3) 1. If a transversal intersects two lines, then the corresponding angles are always equal. State whether true or false and justify your answer. 2. If OA || OB and OA ⊥ OC, then state the relation between OB and OC. 3. If OA || OB and OA || OC, then state the relation between OB and OC.. 4. A line l, intersects two parallel lines m, n at S and R respectively. If the angle between l and n is 97⁰ as shown in the figure, find the remaining angles. 5. In the figure, p is the angular bisector of ∠3 and m, n are parallel lines intersected by l at points A and B respectively. Determine the values of ∠1, ∠2. 6. A transversal l and two parallel lines p, q intersects as shown in the figure. Prove that alternate exterior angles are equal. 7. AB and EF are horizontal lines and BP is a vertical line in the figure. Find the value of ∠ABC, ∠DEF. 57

PRACTICE SHEET - 3 (PS-3)  7. AB and EF are horizontal lines and BP is a verti cal line in the figure. Find the value of ∠ABC, ∠DEF. 8. In the given figure, BE and CF are the angular bisectors of ∠ABC and ∠BCD respectively. Also AB || CD and ∠BCD = 130⁰. Show the two angular bisectors are parallel. 9. Two perpendicular lines are crossed by a pair of parallel lines as shown. Determine the value of ∠1. 58

PRACTICE SHEET - 4 (PS-4) 1. State whether the following statements are true or false. i)  The sum of angles of ∆ABC is 210⁰. ii) In a equilateral, the exterior angle is always 120⁰. 2. In the given figure, Find all the angles. Name the type of ∆BDC. 3. Determine all the angles in the triangles. 4. In the given figure, ∆BDE is a isosceles triangle and a right angled triangle. Find all the angles in the figure. 59

PRACTICE SHEET - 5 (PS-5) I. Choose the correct option. 1. An angle is 32° more than its complimentary angle, then angle is: (A) 52° (B) 29° (C) 58° (D) 26° 2. In the adjoining figure, if l || m and n be the transversal, then the relation between ∠1 and ∠2 is: (A) ∠1 + ∠2 = 180° (B) ∠1 = ∠2 (C) ∠1 - ∠2 = 180° (D) ∠1 + ∠2 = 90° 3. Calculate the value of x. (A) 10° (B) 180° (C) 15° (D) 18° 4. Find ∠CAB. (A) 50° (B) 70° (C) 55° (D) 75° 5. In figure, if l || n and x : y = 3 : 2, then z = (A) 112° (B) 108° (C) 125° (D) 154° 6. If p||q, then value of x is: (A) 70° (B) 120° (C) 60° (D) Cannot be determined 60

PRACTICE SHEET - 5 (PS-5) 7. In figure, lines XY and MN intersect at O. If ∠POY = 70° and x : y = 3 : 2, find z. (A) 136° (B) 95° (C) 70° (D) 120° 8. In adjoining figure if ∠A = (3x + 2°), ∠B = (x - 3°), ∠ACD = 127°, then ∠A= (A) 24° (B) 96° (C) 32° (D) 98° 9. In figure, lines AB and CD intersect at O. If ∠AOC + ∠BOE = 100° and ∠BOD = 60°, find ∠BOE and reflex ∠COE respectively. (A) 30°, 260° (B) 40°, 280° (C) 40°, 260° (D) 30°, 250° 10. Which one of the following statement is true? (A) If two angles forming a linear pair, then each of these angle is of measure 90° (B) Angles forming a linear pair can both be acute angles (C) Both of the angles forming a linear pair can be obtuse angles. (D) Bisectors of the adjacent angles forming a linear pair form a right angle. II. Short answer questions. 1. In the given figure, AB||CD. Find the value of x. 2. The sides BA and DC of the parallelogram ABCD are produced as shown in figure. Prove that a + b+ = x + y. 61

PRACTICE SHEET - 5 (PS-5) 3. In the figure, AB||CD and PQ||RS, find the value of angles 1, 2, 3, 4, 5, 6 and 7. III. Long answer questions. 1. In the given figure, 2b – a = 65° and ∠BOC = 90°, find the measure of ∠AOB, ∠AOD and ∠COD. 2. In figure, the sides AB and AC of ∆ABC are produced to points D and E respectively. If bisectors BP and CP of ∠CBD and ∠BCE respectively meet at point P, then find ∠BPC. 62

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. ∠OAB and 30⁰ are supplementary angles. Find 4. In the given figure, find the angles x, y, z.  ∠OAB. (2 Marks)  (1 Mark) 2. State whether true or false and justify the statement. “If two exterior angles of a triangle are equal, then it is an isosceles triangle.” (1 Mark) 5. Show that a pair of exterior angles on same side of transversal is supplementary if the lines are parallel.  (3 Marks) 3. Two lines intersect at a point P as shown in the figure. If ∠APC=∠APD+60°, find the values of all angles. (3 Marks) 63

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 6. If the four lines m, n, p and q are parallel to 7. In the figure given, show that (3 Marks) one another then find the value of ∠1, ∠2, ∠3 ∠1 + ∠2 + ∠3 = 360⁰. and ∠4.  (2 Marks)  64

7. Triangles Learning Outcomes • Use inequality principles and determine the relationship between lengths and angles in a By the end of this chapter, a student will be able to: triangle. • Verify the congruency of triangles using rules for congruency. • Determine the unknown parameters in any given triangle. Concept Map Key Points o Based on sides:  Scalene triangles: 3 sides are of different • A triangle is a simple closed curve made of three lengths. line segments. It has three vertices, three sides and  Isosceles triangle: 2 sides are of equal length three angles. A triangle is denoted with the symbol,  Equilateral triangle: 3 sides are of equal ‘∆’ length Example: A triangle ABC is denoted as ∆ABC Sides are AB,BC and AC o Based on their angles  Acute angled triangle: All angles are acute. Angles are ∠BAC ,∠ABC and ∠BCA  Right angled triangle: One angle is a right angle (90⁰) Vertices are A, B and C.  Obtuse angled triangle: One angle is an obtuse angle. • Triangles are classified based on their sides and angles. 65

7. Triangles • A line segment joining the midpoint of a side of to the opposite vertex of the triangle. A triangle can triangle and its opposite vertex is called median of have 3 different altitudes based on which side the a triangle. A triangle can have three medians. altitude is measured. Example: In the figure AP, BQ and CR are medians. Example: In ∆ABC, AP is the altitude on the base BC, P, Q and R are midpoints of the sides BC, CA and AB BQ is the altitude on base AC and CR is altitude on respectively. base AB. • In any triangle, sum of lengths of any two sides is greater than the length of the third side. • The sum of the three angles of a triangle is 180⁰. • In a right angled triangle, square of the hypotenuse is equal to the sum of square of the remaining two sides. • If two line segments have same length they are congruent. • If two angles have the same measure they are congruent. • Two triangles are congruent if they are copies of each other and when superposed, they cover each other exactly. Two triangles are congruent if their corresponding parts (angles and sides) that match one another are equal. • Altitude is the perpendicular distance from a side Example: In ∆ABC and ∆XYZ are congruent as shown. Corresponding vertices A & X, B & Y, C & Z. Corresponding sides: AB & XY, BC & YZ, AC & XZ Corresponding angles: ∠A & ∠X, ∠B & ∠Y, ∠C & ∠Z • The symbol used to denote congruency of two elements is ≅. 66

7. Triangles • Theorems, Axioms, Rules Axiom 7.1 SAS Congruence Rule (SAS – Side Angle Side) Two triangles are congruent if two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle. (Included angle means the angle formed by a pair of In sides) ∆ABC, ∆PQR, A & P, B & Q, C & R are assumed to be corresponding vertices and ∆ABC ≅ ∆PQR by SAS rule if any one of the following is satisfied. i) AB = PQ, AC = PR, ∠A = ∠P ii) BC = QR, AB = PQ, ∠B = ∠Q iii) AC = PR, BC = QR, ∠C = ∠R Theorem 7.1 ASA Congruence Rule (ASA – Angle Side Angle) Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of other triangle. (Included sides of two angles means the two angles In ∆ABC, ∆PQR, A & P, B & Q, C & R are assumed to are formed at the end points of the line.) corresponding vertices and AAS Congruence Rule: ∆ABC ≅ ∆PQR by ASA Rule if any one of the following (AAS – Angle Angle Side) is satisfied. Two triangles are congruent if any two pairs of angles i) AB = PQ, ∠A = ∠P, ∠B = ∠Q and one pair of corresponding sides are equal. ii) BC = QR, ∠B = ∠Q, ∠C = ∠R iii) AC = PR, ∠A = ∠P, ∠C = ∠R 67

7. Triangles Theorem 7.4 SSS Congruence Rule (SSS – Side Side Side) If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent. In ∆ABC, ∆PQR, A & P, B & Q, C & R are assumed to corresponding vertices and ∆ABC ≅ ∆PQR by SSS rule then AB = PQ, AC = PR, BC = QR Theorem 7.5 RHS Congruence Rule (RHS – Right angle Hypotenuse Side) If in two right triangles, the hypotenuse and one side of a triangle are equal to hypotenuse and one side of the other triangle, then the two triangles are congruent. In ∆ABC, ∆PQR, A & P, B & Q, C & R are assumed to corresponding vertices and ∆ABC ≅ ∆PQR by RHS rule if any one of the following is satisfied. i) AB = PQ, AC = PR ii) BC = QR, AC = PR • Properties of Triangle Theorem 7.2 Angles opposite to equal sides of an isosceles triangle are equal. Theorem 7.3 In isosceles triangle ABC, The sides opposite to equal angles of an isosceles AB = AC (equal sides) triangle are equal. ∠ABC = ∠ACB 68

7. Triangles Inequalities in Triangle Theorem 7.6 If two sides of a triangle are unequal, the angle opposite to the longer side is larger (or greater). Theorem 7.7 In any triangle, the side opposite to the larger (greater) angle is longer. In ∆ABC, ∠ABC is largest AC is largest Theorem 7.8 The sum of any two sides of a triangle is greater than the third side. In ∆ABC, the three conditions must be satisfied i) AB + BC > AC ii) AB + AC > BC iii) AC + BC > AB Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Pre-requisites • Concept of Congruence PS – 1 • Different types of triangles PS – 2 Triangles • Criteria for Congruence (SAS, ASA, SSS rule, RHS rule) PS – 3 Worksheet for \"Triangles\" • Properties of Isosceles Triangles PS - 4 Evaluation with Self Check or • Inequalities in Triangles. PS - 5 Peer Check* Self Evaluation Sheet ---- 69

PRACTICE SHEET - 1 (PS-1) 1. Name the type of triangles based on its angles 7. Identity the equal sides and equal angles of the and sides: given isosceles triangle. 2. Can a right angled triangle also be Isosceles triangle? 3. Find the lengths AP and PB, if CP is the median of the right angled triangle. 4. Check if the triangles shown in the figure are congruent. 5. Among the given triangles which are congruent: In case the triangles are congruent, mention the corresponding sides that are equal and list the corresponding elements in the congruent triangles. 6. Locate the following elements in the given a right angled triangle. i) Vertex where the right angle is present ii) Hypotenuse 70

PRACTICE SHEET - 2 (PS-2) 1. Identify the following elements from the given figure. i) Included angle for AB & AC ii) Included angle for BA and CB iii) Included side for ∠B and ∠C iv) Included side for ∠ACB & ∠CAB 7. A shopkeeper tells Ravi that if he could help in cutting a rectangular cake into 4 equal triangles such that one side of the triangle should be equal to AB, then he will give Ravi one piece of cake for free. 2. If the corresponding angles of two triangles are equal, will the two triangles be always be Congruent? Give example. Ans: No. 8. In the given figure, AB || CD and AD = BC. Prove that ∆ABC ≅ ∆ABD. 3. Can any two equilateral triangles be congruent? Give reason. 4. If ∆ABC ≅ ∆PQR and ∆ABC ≅ ∆XYZ, then show that ∆PQR ≅ ∆ XYZ. 5. In a quadrilateral ABCD, the diagonals intersect at point A as shown in the figure. The triangles ABC and AED are equilateral triangles. Show that i) ∆ABE ≅ ∆ACD  ii) ∆BEC ≅ ∆BDC   iii) ∆DEC ≅ ∆DBE 6. Two congruent scalene triangles ABO and OCD are created on a line segment AD as shown. Show that ∆OAB ≅ ∆OBC. 71

PRACTICE SHEET - 32 (PS-32) 1. If the length of equal sides of an isosceles triangle is equal to the length of equal sides of another isosceles triangle, by which congruence rule will the two triangles be congruent? 2. Can the angle opposite to the equal sides of the isosceles triangle be an obtuse angle? 3. Can the hypotenuse of the right angle triangle, be an equal side of isosceles triangle? 4. State whether the following statements are true or false. Give reasons. i) A right angled triangle can also be isosceles triangle. ii) Equilateral triangle is an isosceles triangle. 5. In an isosceles triangle ∆ABC, AB = AC and AD ⊥ BC. Show that AD is the angular bisector of ∠BAC. 6. ABC is an isosceles triangle with sides AB = BC and CD is the median drawn as shown. Check whether the ∆ACD and ∆DCB are congruent. 7. In the figure below, AB = AD, BC = CD and diagonal AC bisects BD. Show that i) ∆ABC ≅ ∆ADC ii) ∆AOB ≅ ∆AOD iii) ∆BOC ≅ ∆DOC iv) OA ⊥ BD 72

PRACTICE SHEET - 24 (PS-24) 1. Why is the length of the side opposite to the obtuse angle in an obtuse-angled triangle always the largest angle in the triangle? 2. Can the sides of the triangle be in the ratio i) 1: 2: 3 ii) 1.5 : 2 : 3 3. In the given triangle arrange angles in the descending order. 4. Show that in a right angle triangle ABD as shown in the figure, ∠BAC < ∠CAD if BC = CD = AD. 5. Two triangles are constructed on a line segment AB such that the third points lies on the perpendicular bisector of AB. Show that if the sum of sides of the triangles increases, the included angle reduces. 73

PRACTICE SHEET - 5 (PS-5) I. Choose the correct option. 1. Which of the following is not a criterion for congruence of triangles? (A) SSS (B) SAS (C) ASA (D) SSA 2. In a ∆ABC, AB = 10 cm, AC = 10 cm and ∠A = 80°, then ∠B = (A) 80° (B) 180° (C) 50° (D) 40° 3. In triangles ABC and RQP, if AB = AC, ∠C = ∠P and ∠B = ∠Q, then two triangles are: (A) isosceles but not necessarily congruent (B) congruent but not isosceles (C) isosceles and congruent (D) neither congruent nor isosceles 4. In the given figure, PS is the median, bisecting angle P, then ∠QPS is: (A) 110° (B) 45° (C) 70° (D) 65° 5. In the given figure, PQR is an equilateral triangle and QRST is a square. Then ∠PSR = (A) 48° (B) 30° (C) 84° (D) 36° 6. The sum of altitudes of a triangle is _______ than the perimeter of the triangle. (A) greater (B) equal (C) half (D) less 7. Observe the figure, AD is the median of ∆ABC, BL and CM are perpendiculars drawn, from B and C re- spectively. Select the correct option ((A) BL||CM (B) ∆BLD ≅ ∆CMD (C) LD = AL (D) All of these 8. In the given figure, AB ⊥ BE and EF ⊥ BE. Also, BC = DE and AB = EF. Then (A) ∆ABD ≅ ∆FEC (B) ∆ABD ≅ ∆EFC (C) ∆ABD ≅ ∆CMD (D) ∆ABD ≅ ∆CEF 9. Which of the following is a correct statement? (A) Two triangles having same shape are congruent. (B) If two sides of a triangle are equal to the corresponding sides of another triangle, then two triangles are congruent. (C)If the hypotenuse and one side of one right triangle are equal to the hypotenuse and one side of the other triangle, then the triangles are not congruent. (D) None of these. 10. ∆ABC and ∆PBC are two isosceles triangles on the same base BC and vertices A and P are on the same side of BC. A and P are joined, then (A) ∆ABP ≅ ∆ACP (B) AP bisects ∠A (C) ∠ABC = ∠ACP (D) Both (A) and (B) 74

PRACTICE SHEET - 5 (PS-5) II. Short answer questions. 1. In the figure, AB= AC, ∠ACM = 140° and ∠PAB = x. Find the value of x. 2. The angles of a triangle are (x - 40°), (x - 20°) and  1 x -10°  . Find the products of the digits of x.  2  3. In the figure, show that 2(AC + BD) > AB + BC + CA + DA III. Long answer questions. 1. In figure, if lines PQ and RS intersect at a point T such that ∠PRT = 40°, ∠RPT = 95° and ∠TSQ = 75°, ∠ SQT = K × 60°. Find the value of K. 2. The image of an object placed at a point A before a plane mirror LM is seen at the point B by an observer at D as shown in figure. Prove that the image is far behind the mirror as the object is in front of the mirror. 75

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. In a right angled triangle, can the hypotenuse are equal to three angles and one side of be equal to any other side?  (1 Mark) another triangle, then the two triangles will be congruent.  (2 Marks) 2. In a triangle, if ∠A > ∠B > ∠C what is the relation between the sides?  (1 Mark) 5. In the given figure, EC and EA are the angular bisectors of the external angles of ∆OAC. If OA > OC, find the larger side between EC and EA.  (3 Marks) 3. The hypotenuse of two right angled triangles is equal. Are the triangles congruent?  (1 Mark) 4. State whether the following statements are true or false. Explain. i) In isosceles triangle two angles will always be equal. ii) If three angles and one side of a triangle 76

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 6. Show that the triangle formed by joining the midpoints of the sides of the sides of the equilateral is also an equilateral triangle. (4 Marks) 7. Two perpendicular lines AB and CD intersect at O. Right angled triangles are drawn on these lines as shown. Find the relation between the sizes of the hypotenuses of the 4 triangles if AC > BD.  (3 Marks) 77

8. Quadrilaterals Learning Outcome By the end of this chapter, a student will be able to: • Explain the different properties of a given type of quadrilateral. • Determine whether a quadrilateral is a parallelogram. • Calculate the relation between the various elements of a parallelogram. Concept Map Types of Quadrilaterals Theorems on Parallelograms and their Properties Diagonals of parallelogram Parallelogram Rhombus Opposite side Rectangle Square Opposite angles Trapezium Kite Angles, lengths of Quadrilateral Parallelograms and their Midpoint Angle Sum Property Theorem relationships Key Points • Types of Quadrilaterals • A figure formed by joining four points in order is Trapezium called a quadrilateral. It has four sides, four angles If one pair of opposite and four vertices. sides of a quadrilateral is parallel then it is called a Example: trapezium. Doors, book, black board, windows, etc are all Trapezium is not a ABCD is a trapezium, quadrilaterals. parallelogram. AB || CD • Angle sum Property of a Quadrilateral: The sum of Parallelogram ABCD is a parallelogram, If both pairs of opposite AB = CD, AB || CD angles of a quadrilateral is 360⁰. sides of a quadrilateral AD = BC, AD || BC In the figure, ABCD is a quadrilateral, are parallel, then it is ∠DAC + ∠ABC + ∠BCA + ∠CDA = 360⁰ called a parallelogram. The opposite sides and opposite angles of parallelogram are equal. A parallelogram is a trapezium. 78

8. Quadrilaterals Rectangle • Theorems related to Parallelograms If one of the angles of Theorem 8.1 a parallelogram is a A diagonal of a right angle, then it is a parallelogram rectangle. divides it into two congruent triangles. Rectangle is a ABCD is a rectangle Theorem 8.2 ABCD is parallelogram, AC, BD quadrilateral with four AB = CD, AB || CD, In a parallelogram, are diagonals. right angles and pair AD = BC, AD || BC opposite sides are ∆ABC ≅ ∆ADC of opposite sides being ∠DAB = ∠ABC = ∠BCD = equal. ∆ABD ≅ ∆CBD equal. ∠CDA = 90⁰ A rectangle is not a square. Rhombus In the parallelogram, if the four sides are equal, then it is called a Theorem 8.3 ABCD is parallelogram if rhombus. The diagonals If each pair of AB = CD of rhombus are opposite sides of AD = BC a quadrilateral is perpendicular to each ABCD is a rhombus equal, then it is a and bisect one another. AB = CD = BC = DA parallelogram. Theorem 8.4 Rhombus is not a square. AB || DC BC || DA In a parallelogram, opposite angles are Square equal. In a parallelogram, if all the sides are equal and one angle is a right angle, then it is called a square. Theorem 8.5 ABCD is Parallelogram A square is a rectangle ABCD is a square If in a quadrilateral, ∠DAB = ∠DCB and a rhombus. AB = BC = CD = CA each pair of ∠CDA = ∠CBA Kite AB || CD, AD || BC In a quadrilateral if, two ∠DAB = ∠ABC = ∠BCD = opposite angles is pairs of adjacent angles ∠CDA = 90⁰ equal, then it is a are equal, then it is parallelogram. called a kite. Theorem 8.6 The diagonals of a parallelogram bisect each other. A kite is not a Theorem 8.7 ABCD is parallelogram parallelogram. If the diagonals of a OA = OC quadrilateral bisect OD = OB ABCD is a Kite each other, then it is AB = CD a parallelogram. AD = DC 79

8. Quadrilaterals Theorem 8.8 ABCD is parallelogram A quadrilateral is AB = CD, AB || CD a parallelogram, if AD = BC, AD || BC a pair of opposite sides is equal and parallel. • Midpoint Theorem Theorem 8.9 The line segment joining the midpoint of two sides of a triangle is parallel to the third side. Theorem 8.10 The line drawn through the midpoint of one In ∆PQR side and parallel to A is midpoint of PR another side bisects AB || PQ the third side. B is midpoints of QR Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Pre-requisites • Elements of a Quadrilateral PS – 1 • Different types of Quadrilateral and their properties. • Angle Sum Property PS – 2 Quadrilaterals • Properties of Parallelogram PS – 3 • Another condition for a Quadrilateral to be a Parallelogram • Midpoint Theorem PS - 4 Worksheet for “Quadrilaterals” PS - 5 Evaluation with Self Check or ---- Self Evaluation Sheet Peer Check* 80

PRACTICE SHEET - 1 (PS-1) 1. In the given figure, if AB = 5 cm and AD = 3 cm. If ABCD is a parallelogram, find the value of CD and BC. 2. In a square all angles are equal, can the four sides be parallel to one another? 3. In the quadrilateral ABCD, AB = 5 cm, DC = 3 cm. Find the values of the remaining sides. 4. In the figure shown, AB = BC and AD = CD. Also ∠DAB = ∠BCD = 90⁰. Name the quadrilateral. If AB = 4 cm and AD = 3 CM, find the value of BD. 81

PRACTICE SHEET - 2 (PS-2) 1. If the three angles of a quadrilateral are right angles, what is the value of the 4th angle? 2. Ravi and Suresh play a game of drawing quadrilaterals. After drawing, they pick one angle alternately and the score will be the sum of the values of the selected angles. Higher score means winning the game. If the figure shown is drawn by the two players and Ravi selects ∠P and ∠S, who has won the game. Find the value of ∠R. 3. Check if the following statements are true or false? i) A quadrilateral has 4 acute angles. ii) A quadrilateral has 4 obtuse angles. 4. In a quadrilateral shown in the figure, if the points A, C are joined, then AC bisects the ∠A and ∠C. Find the value of ∠B and ∠D. 5. If the angles of a quadrilateral are in the ratio of 1:2:3: 4, find the angles of the quadrilateral. Is there more than one combination of possible angles? 6. The four angles of the quadrilateral in order are in the ratio of a:b:a:b. State at least two combination of angles if possible. How many different combinations are possible for quadrilaterals? 82

PRACTICE SHEET - 23 (PS-23) 1. In a quadrilateral if the diagonals are equal, what is the nature of the quadrilateral? 2. Name the type of quadrilateral if its diagonals are equal and are at right angles. 3. State whether which of the following statements is true or false. Justify. i) The diagonal of a rhombus divides it into two isosceles triangles. ii) A diagonal of a rhombus in some cases divide it into two equilateral triangles. iii) Diagonals of a square divide it into two isosceles triangles. iv) Diagonals of a square divide it into 2 equilateral triangles. 4. In a quadrilateral, the diagonals divide it into 4 congruent triangles. Is the quadrilateral a rectangle, parallelogram, rhombus or a square. Give reasons? 5. If PQRS is a parallelogram and PR & QS are its diagonals, find at least one angle/side which is equal to the following: i) PT  ii) QX  iii) RT  iv) ∠OQR v) ∠OST vi) ∠OPS  vii) SO  viii) PR 6. In a parallelogram PQRS, M is the midpoint of side RS. If ∆PSM and ∆QRM are congruent, then prove that PQRS is a rectangle. 7. In a parallelogram PQRS, M is the midpoint of side RS. If QR = MR, prove that ∆PMQ is a right angled triangle. 83

PRACTICE SHEET - 3 (PS-3) 8. Two lines AB and CD intersect at their midpoint O. Show that ACBD will be a parallelogram. 9. ABCD is a parallelogram and Q, S are mid points of sides BC, AD respectively. QR and PS are drawn perpendicular to the sides CD and AB respectively. ii) Find whether RS = PQ, DR = PB iii) What is the nature of the quadrilateral PQRS? 84

PRACTICE SHEET - 34 (PS-34) 1. In a triangle, if a line is drawn parallel to any ii) ∆BAS and ∆PQA are congruent. one side, will it bisect the other two sides? iii) QABR is a square, if ∆PRS is an isosceles right 2. In ∆ABC, the midpoint of side AB is point P and angle triangle. points Q, R are on the sides BC and CA of the 7. In a right angled triangle, ∆ACF, ∠CAF = 90⁰. If triangle. If PQRA is a parallelogram, prove that Q and R are also mid points of the sides of the B, D, E are the mid points of the sides then show triangle. that AC = 2ED, AF = 2BD and CF = 2BE. 3. P, Q, R and S are the midpoints of the sides of the quadrilateral ABCD as shown in figure. Show that i)  lines PS and QR are parallel lines ii)  lines PQ and RS are parallel lines 4. PQRS is a quadrilateral in which the midpoints of sides PS and QR are joined by a line XY. Show that if XY is parallel to RS, then XY is parallel to PQ. 5. Show that lines joining the mid points of the triangle divide the triangle into 4 equal parts. 6. Q, B, A are the mid points of the three sides of the right angle triangle, ∆PRS. Show that i) QABR is a rectangle. 85

PRACTICE SHEET - 5 (PS-5) I. Choose the correct option. 1. PQRS is a square, PR and SQ intersect at O. The measure of ∠POQ is: (A) 45° (B) 180° (C) 90° (D) None of these 2. If a quadrilateral has two adjacent sides are equal and the opposite sides are unequal, then it is called a: (A) Kite (B) parallelogram (C) square (D) rectangle 3. The two diagonals are equal in a: (A) parallelogram (B) rhombus (C) trapezium (D) rectangle 4. One of the diagonals of a rhombus is equal to a side of the rhombus. The pair of unequal angles of the rhombus are: (A) 60°, 120° (B) 60°, 80° (C) 120°, 120° (D) 100°, 120° 5. In a square ABCD, AB = (2x + 3) cm and BC = (3x – 5) cm. Then, the value of x is: (A) 5 (B) 10 (C) 8 (D) 11 Ans: C 6. A quadrilateral has three acute angles each measuring 70°. The measure of fourth angle is: (A) 140° (B) 150° (C) 105° (D) 180° 7. Which is not correct about rectangle EFGH? (A) ∠E = ∠F = ∠G = ∠H = 90° (B) EG = FH (C) EF = GH AND HE = FG (D) EG and FH are ⊥ bisectors 8. The length and breadth of a rectangle are in the ratio 4 : 3. If the diagonal measures 2cm, then the perim- eter of the rectangle is: (A) 58 cm (B) 70 cm (C) 60 cm (D) 10 cm 9. If an angle of a parallelogram is two-third of its adjacent angle, then the angles of a parallelogram are: (A) 37°, 143°, 37°, 143° (B) 108°, 72°, 108°, 72° (C) 68°, 112° , 68°, 112° (D) None of these 10. Which of the following is true? (A) In a parallelogram, the diagonals bisect each other. (B) In a parallelogram, the diagonals are equal. (C) In a parallelogram, the diagonals intersect each other at right angles. (D) In any quadrilateral, if a pair of opposite sides are equal, it is parallelogram. II. Short answer questions. 1. In the given figure, ABCD is a parallelogram in which ∠DAB = 75° and ∠DCB = 60°. Compute ∠CDB and ∠ADB. 2. Prove that all sides of a rhombus are equal, if diagonals bisect each other at 90°. 3. If the bisectors of two adjacent angles A and B of a quadrilateral ABCD intersect at a point O such that ∠C + ∠D = k ∠AOB, then the value k – 1 is: 86