202110219-TRIUMPH-STUDENT-WORKBOOK-MATHEMATICS-G07-PART1

Long Answer Type Questions 12 [AS1] Divide: 8.12 by 2.5 13 [AS4] Vamsi has to pay Rs. 50 a month for 500 text messages. For each message over 500, he has to pay an extra Rs. 0.55. How much does Vamsi owe if he sent 768 text messages in a month? 14 [AS4] Each of the six buckets of equal capacity holds 120.06 litres of water. How many litres of water can be stored in 12 such buckets? 15 [AS4] A green club is planting trees. Each tree they plant costs them Rs. 25.50. If they have Rs. 510, how many trees can they plant? EXERCISE 2.6. MULTIPLICATION OF DECIMAL NUMBERS 47

EXERCISE 2.7 INTRODUCTION TO RATIONAL NUMBERS – POSITIVE FRACTIONAL NUMBERS 2.7.1 Key Concepts i. The numbers in the form p where p, q are integers such that 'q' is not equal to zero are called q rational numbers. ii. The set of rational numbers is represented by Q. iii. Q includes all integers, positive fractional numbers and negative fractional numbers. iv. All rational numbers can be represented on a number line. 2.7.2 Additional Questions Objective Questions 1. [AS1] One of the rational numbers between 3 and 4 is . (A) 10 (B) 10 2 3 (C) 10 (D) 10 4 5 2. [AS1] The rational number 19 lies between . (B) 19, 20 9 (D)2, 3 (A) 1, 2 (C)9, 10 3. [AS1] Among the following rational numbers, 7 , 3 , 4 and 8 , the greatest one is . 9 4 5 9 (A) 4 (B) 8 5 9 (C) 7 (D) 3 9 4 4. [AS1] The smallest among 6 , 7 , 3 and 8 is . 9 4 4 9 (A) 6 (B) 7 9 4 (C) 3 (D) 8 4 9 EXERCISE 2.7. INTRODUCTION TO RATIONAL NUMBERS –POSITIVE FRAC. . . 48

5. [AS1] There are rational numbers between any two integers. (A) 0 (B) 1 (C) 2 (D)Infinitely many Very Short Answer Type Questions 6 [AS1] State true or false. (i) Every positive integer is a positive rational number. (ii) Zero is a negative rational number. [ ] [ ] (iii) 3 + 1 is the rational number between 2 and 3. [ ] 2 4 [ ] [ ] (iv) There are no rational numbers between 2 and 3. (v) There are no positive fractions between 2 and 4 . 5 5 7 [AS2] Fill in the blanks. (i) −5 11 9 −16 (ii) −3 4 8 (−7) (iii) 3 11 4 8 (iv) 5 −7 −14 12 (v) −7 5 12 12 EXERCISE 2.7. INTRODUCTION TO RATIONAL NUMBERS –POSITIVE FRAC. . . 49

Short Answer Type Questions 8(i) [AS1] Write two equivalent rational numbers for the following. a) 3 b) −6 5 7 (ii) [AS1] Write two equivalent rational numbers for the following. a) 7 b) 8 c) 10 −10 15 13 9(i) [AS2] Compare the following pairs of rational numbers: 7 and 9 15 20 (ii) [AS2] Compare the following pairs of rational numbers: 5 and −7 −14 21 10 [AS5] Represent three rational numbers between the two integers 4 and 5 on the number line. Long Answer Type Questions 11 [AS1] Between which two numbers do 5 2 , 2 4 , 1 1 and 6 lie on the number line? 3 5 7 7 12 [AS1] Write 5 positive fractions between 2 and 11 . 12 12 13 [AS5] Represent the following numbers on the number line. (i) 1 (ii)23 (iii) −4 (iv) 2 1 (v) −1 3 3 3 3 4 EXERCISE 2.7. INTRODUCTION TO RATIONAL NUMBERS –POSITIVE FRAC. . . 50

CHAPTER 3 SIMPLE EQUATIONS EXERCISE 3.1 INTRODUCTION 3.1.1 Key Concepts i. Simple equations help in solving various problems in daily life. ii. For balancing an equation we a. add the same number on both the sides or b. subtract the same number from both the sides or c. multiply both the sides by the same number or d. divide both the sides by the same number, so that the equality remains undisturbed. iii. An equation remains the same if the LHS and the RHS are interchanged. 3.1.2 Additional Questions Objective Questions 1. [AS1] The L.H.S of the equation 5x − 19 = 3x + 7 is . (A) 5x (B) 5x –19 (C) 3x (D)3x + 7 2. [AS1] The solution of 3x − 4 = 8 is x = . (A) 4 (B) 32 3 3 (C) 3 (D) 4 EXERCISE 3.1. INTRODUCTION 51

3. [AS1] The R.H.S of the equation 3x − 5 = 7 is . 2 (A) 7 (B) − 5 (C) 3x (D) 3x − 5 2 2 4. [AS1] The L.H.S of the equation 2x − 3 = 7x + 5 is . (A) 2x (B) 7x (C)2x − 3 (D)7x + 5 5. [AS1] The R.H.S of the equation 35 = 7x − 20 is . (A) 7x (B) 7x − 20 (C) 35 (D) None of these Very Short Answer Type Questions 6 [AS1] Answer the following questions in one sentence. (i) State the L.H.S and R.H.S: 7x – 2 = 2 (ii) State the L.H.S and R.H.S: 3y = 5 (iii) State the L.H.S and R.H.S: 12x + 3 = 39 (iv) State the L.H.S and R.H.S: 16p = 64 (v) State the L.H.S and R.H.S: 25 = 2q + 3 Short Answer Type Questions 7(i) [AS1] Solve by trial and error method: 3x – 5 = 10 (ii) [AS1] Solve by trial and error method: – 4p = 16 EXERCISE 3.1. INTRODUCTION 52

EXERCISE 3.2 EQUATIONS–SOLVING THE EQUATION 3.2.1 Key Concepts i. To solve a simple equation, we transpose a term from one side to the other. ii. While transposing terms from one side to another a. ‘+’ quantity becomes ‘–’ quantity b. ‘–’ quantity becomes ‘+’ quantity c. ‘×’ quantity becomes ‘÷’ quantity d. ‘÷’ quantity becomes ‘×’ quantity (i.e.,) when the terms are transposed they get opposite signs and the term which multiplies one side, divides the other side. 3.2.2 Additional Questions Objective Questions 1. [AS1] 4x − 9 = 9x + 11 then x = . (A) 4 (B) − 4 (C) 20 (D)− 20 2. [AS1] The value of y which satisfies 6y − 17 = 10 is . (A) 27 (B) − 33 (C) 33 (D) 9 2 EXERCISE 3.2. EQUATIONS–SOLVING THE EQUATION 53

3. [AS1] The value of 'p' satisfying 3p + 21 = 8 is . 4 (A) − 13 (B) 13 (C) −52 (D) 52 3 3 4. [AS1] The solution of 7x = 21 is x = . (A) 3 (B) 28 (C) 14 (D) 147 5. [AS1] The solution of 4x − 3 = 3x + 4 is . (A) 3 (B) 4 (C) 12 (D) 7 Short Answer Type Questions 6(i) [AS1] Solve the equation without transposing: 4x – 3 = 7 (ii) [AS1] Solve the equation without transposing: – 7y = 49 7(i) [AS1] Solve and check the result: 6x + 2 = 20 (ii) [AS1] Solve and check the result: 7y – 5 = 30 Long Answer Type Questions 8 [AS1] Solve the equation without transposing: 3x+2 = 5 7 9 [AS1] Solve and check the result: 5 p−2 = 24 2 EXERCISE 3.2. EQUATIONS–SOLVING THE EQUATION 54

EXERCISE 3.3 USAGE OF ALGEBRAIC EQUATIONS IN SOLVING DAY TO DAY PROBLEMS 3.3.1 Key Concepts i. Simple equations help in solving various problems in daily life. ii. An equation remains the same if the LHS and the RHS are interchanged. 3.3.2 Additional Questions Objective Questions 1. [AS1] The product of two numbers is 153. If one of the numbers is 9, then the other number is . (A) 162 (B) 144 (C) 17 9 (D) 153 2. [AS1] The sum of four numbers is 229. If the sum of three of them is 193, then the fourth number is . (A) 422 (B) 229 193 (C) 36 (D)229 × 193 3. [AS1] The value of 'm' in 24m + 13 = 11m + 182 is . (A) 13 (B) 195 (C) 169 (D) 169 35 4. [AS1] The value of 'p' in 71 − 7p = 3 − 3p is . (B) 17 (A) 74 (D)−17 10 (C) 37 2 EXERCISE 3.3. USAGE OF ALGEBRAIC EQUATIONS IN SOLVING DAY TO . . . 55

5. [AS1] If the sum of three consecutive numbers is 225, then the middle number is . (A) 74 (B) 76 (C) 70 (D) 75 Short Answer Type Questions 6(i) [AS1] If the sum of two numbers is 72 and one of the numbers is 23, find the other. (ii) [AS1] If the sum of three consecutive numbers is 63, find them. 7(i) [AS1] Find the value of x in 12x + 5 = 125. (ii) [AS1] Find the value of 'p' in 18p – 7 = 47. Long Answer Type Questions 8 [AS1] (i) The present age of a mother is thrice that of her daughter. Four years ago, the daughter’s age was 11 years. Find the present age of the mother. (ii) Rene is 6 years older than her younger sister. After 10 years, the sum of their ages would be 50 years. Find their present ages. 9 [AS1] (i) The cost of two tables and three chairs is Rs. 705. If a table costs Rs. 40 more than a chair, find the cost of a table and a chair. (ii) Manisha is 5 years younger than Meghana. Four years later, Meghana will be twice as old asManisha. Find their present ages. 10 [AS1] (i) Find the value of x from the following figure: (ii) In a class of 42 students, the number of boys is 2 of the number of girls. Find the number of boys 5 and girls in the class. EXERCISE 3.3. USAGE OF ALGEBRAIC EQUATIONS IN SOLVING DAY TO . . . 56

CHAPTER 4 LINES AND ANGLES EXERCISE 4.1 INTRODUCTION 4.1.1 Key Concepts i. Points have no dimensions. They denote locations in a three-dimensional space. ii. A line segment is the distance between two points. iii. When a line segment is extended to infinity only on one side, it is called a ray. iv. When a line segment is extended to infinity on both the sides, it is called a line. 4.1.2 Additional Questions Objective Questions . 1. [AS3] A simple closed figure drawn using four line segments is called (A) A triangle (B) A quadrilateral (C)A pentagon (D) None of these 2. [AS3] An angle measure of more than 90◦ is known as angle. (A) An acute (B) A right (C)An obtuse (D)A straight 3. [AS3] An angle whose measure lies between 180◦ and 360◦ is known as angle. (A) A reflex (B) A straight (C)An obtuse (D)A right EXERCISE 4.1. INTRODUCTION 57

4. [AS3] The measure of a right angle is . (A) 0◦ (B) 60◦ (C) 180◦ (D) 90◦ 5. [AS3] The measure of a straight angle is . (A) 0◦ (B) 60◦ (C) 180◦ (D) 90◦ Very Short Answer Type Questions . 6 [AS1] Fill in the blanks. . (i) A line which has two end points is called a (ii) A line which has only one end point is called a . (iii) A __________________ has no end points. (iv) A ray is a part of a . (v) The given ray is represented as 7 [AS2] Answer the following questions in one sentence. (i) Write any five examples of angles that you have observed in your surroundings. (E.g.: The angle formed when a pair of scissors is opened.) (ii) The measure of an angle is 86◦. What type of angle is it? (iii) The measure of an angle is greater than 90◦ and less than 180◦. Name the type of angle. 8 [AS2] Choose the correct answer. . (i) The two rails of a railway track are an example of (A) Parallel lines (B) Intersecting lines (C)Perpendicular lines (D) None of these EXERCISE 4.1. INTRODUCTION 58

(ii) An example of perpendicular lines is . (A) The two rails of a railway track. (B) The two edges representing length and height of a room. (C)The floor and roof of a room. (D)None of these (iii) In the figure, the lines l and n are lines. (A) Parallel (C) Intersecting (B) Perpendicular (D) Coincident (iv) In the given figure, l and m are lines. (B) Perpendicular (A) Parallel . (D) Coincident (C) Vertical (B) 0◦ (v) The angle between parallel lines is (D) 360◦ (A) 180◦ (C) 90◦ EXERCISE 4.1. INTRODUCTION 59

9 [AS5] Answer the following questions in one sentence. (i) What type of angle is ∠ABC? (ii) Identify acute and obtuse angles from the figure. Short Answer Type Questions 10 [AS3] Illustrate the following: (i) CD −−→ (iii) Point X (ii) AB (iv) ←P→Q (v) −−−→ MN 11 [AS5] Draw the figures for the following: (i)−P−→Q (ii)R←→S (iii) AB (iv) ∠ABC 12(i) [AS5] Name all the different line segments in the figure: EXERCISE 4.1. INTRODUCTION 60

(ii) [AS5] Name the line segments in these figures. a) b) 13(i) [AS5] Identify the following angles as acute, obtuse or right. (ii) [AS5] Draw one of each kind: a) Acute angle b) Obtuse angle c) Right angle EXERCISE 4.1. INTRODUCTION 61

Long Answer Type Questions 14 [AS5] Name the figures drawn. EXERCISE 4.1. INTRODUCTION 62

EXERCISE 4.2 COMPLEMENTARY ANGLES 4.2.1 Key Concepts i. Complementary angles: If the sum of two angles is 90◦, the angles are called complementary angles. e.g: 55◦ and 35◦ are complementary angles. 4.2.2 Additional Questions Objective Questions 1. [AS1] If one of the two complementary angles is 67 ◦, then the second angle is . (A) 113◦ (B) 33◦ (C) 293◦ (D) 23◦ 2. [AS1] If two complementary angles are equal in magnitude then each of the angles is . (A) 45◦ (B) 60◦ (C) 90◦ (D) 180◦ 3. [AS1] If two complementary angles are in the ratio of 1: 2 then, the angles in order are . (A) 60◦, 30◦ (B) 30◦, 60◦ (C)45◦, 45◦ (D)90◦, 0◦ 4. [AS1] If one of the complementary angles is three times the other, then the angles are . (A) 15◦, 75◦ (B) 45◦, 45◦ (C) 22 1 ◦ , 67 1 ◦ 2 2 (D)90◦, 0◦ EXERCISE 4.2. COMPLEMENTARY ANGLES 63

5. [AS3] The sum of two complementary angles is . (A) 0◦ (B) 180◦ (C) 90◦ (D) 360◦ Short Answer Type Questions 6(i) [AS1] Two angles are complementary to each other and one of them is twice the other. Find them. (ii) [AS1] Find the complementary angles of the following. a) 40◦ b) 79◦ c) 55◦ 7 [AS2] Ravi says, “No angle in a pair of complementary angles can be obtuse”. Do you agree? Give reason. EXERCISE 4.2. COMPLEMENTARY ANGLES 64

EXERCISE 4.3 SUPPLEMENTARY ANGLES 4.3.1 Key Concepts i. Supplementary angles: Two angles are said to be supplementary if their sum is 180◦. e. g: (100◦, 80◦), (110◦, 70◦), (60◦, 120◦) ii. Angles in a pair of supplementary angles may be either acute or right or obtuse. iii. Two right angles are always supplementary. 4.3.2 Additional Questions Objective Questions 1. [AS1] If one of the two supplementary angles is 67 ◦ , then the other angle is . (A) 113◦ (B) 33◦ (C) 293◦ (D) 23◦ 2. [AS1] The supplement of 132◦ is . (A) 148◦ (C) 48◦ (B) 18◦ (D) 28◦ 3. [AS1] If the two angles in a pair of supplementary angles are in the ratio of 3 : 2, then the angles in order are . (A) 108◦, 72◦ (B) 72◦, 108◦ (C)90◦, 90◦ (D)60◦, 120◦ 4. [AS1] If an angle in a pair of supplementary angles is 8 times the other, then the two angles in decreasing order are . (A) 20◦, 160◦ (B) 40◦, 140◦ (C)140◦, 40◦ (D)160◦, 20◦ EXERCISE 4.3. SUPPLEMENTARY ANGLES 65

5. [AS3] If the sum of two angles is 180◦, then they are called angles. (A) Complementary (B) Supplementary (C) Straight (D) None of these Short Answer Type Questions 6(i) [AS1] What are supplementary angles? Two angles are equal and supplementary to each other. Find them. (ii) [AS1] Find the supplementary angles of the following angles: a) 85◦ b) 115◦ c) 128◦ Long Answer Type Questions 7 [AS2] Two acute angles cannot form a pair of supplementary angles. Justify. EXERCISE 4.3. SUPPLEMENTARY ANGLES 66

EXERCISE 4.4 ADJACENT ANGLES 4.4.1 Key Concepts i. Adjacent angles: Two angles with a common vertex and a common arm are called adjacent angles. The non–common arms lie on either sides of the common arm. e. g: ∠AOB and ∠BOC are adjacent angles. 4.4.2 Additional Questions Objective Questions angles. 1. [AS1] If two adjacent angles are 90 ◦ each, they form (A) Acute (B) Obtuse (C) Straight (D) None of these 2. [AS1] A pair of adjacent angles in the figure given is ______. (A) ∠P, ∠R (B) ∠Q, ∠S (C)∠P, ∠Q (D) None of these 3. [AS1] A pair of non-adjacent angles in the figure given is_______. (A) ∠P, ∠R (B) ∠Q, ∠S (C)∠P, ∠Q (D)Both A and B 4. [AS1] In the given triangle, there are pairs of adjacent angles. (A) 0 (B) 1 (C) 2 (D) 3 EXERCISE 4.4. ADJACENT ANGLES 67

5. [AS1] The adjacent angles in a parallelogram are . (A) Complementary (B) Supplementary (C)Right angles (D)None of these Very Short Answer Type Questions 6 [AS5] Answer the following questions in one sentence. (i) In the given figure, 68 a) How many pairs of adjacent angles are formed? b) Why are these angles called adjacent angles? c) Does this figure contain any opposite angles? d) What is the point of intersection? (ii) Name the adjacent angles in the following figures. a) b) c) EXERCISE 4.4. ADJACENT ANGLES

d) e) Short Answer Type Questions 7 (i) [AS5] a) Can two adjacent angles be supplementary? Draw a figure. b) Can two adjacent angles be complementary? Draw a figure. (ii) [AS5] Draw one of each kind. a) Adjacent angles b) Right–angled adjacent angles c) Non–adjacent angles Long Answer Type Questions 8 [AS4] Give five examples of adjacent angles which you observe in your daily life. EXERCISE 4.4. ADJACENT ANGLES 69

EXERCISE 4.5 LINEAR PAIR 4.5.1 Key Concepts i. Linear pair of angles: When a ray stands on a line, the pair of angles thus formed are called linear pair of angles and their sum is 180◦. In the figure, ∠X and ∠Y are called linear pair of angles. Linear pair of angles are always adjacent. 4.5.2 Additional Questions Objective Questions 1. [AS1] The sum of two angles forming a linear pair is . (A) 0◦ (B) 60◦ (C) 90◦ (D) 180◦ 2. [AS1] If one of the angles of a linear pair is 125◦, then the other angle is . (A) 55 ◦ (B) 235◦ (C) 145◦ (D) 90◦ 3. [AS1] If the two angles in a linear pair are in the ratio of 5 : 4, then the angles in order are . (A) 80◦, 100◦ (B) 100◦, 80◦ (C)90◦, 90◦ (D) None of these 4. [AS1] One of the two angles in a linear pair is 5 times the other. Then the two angles in decreasing order are . (A) 120◦, 60◦ (B) 60◦, 120◦ (C)150◦, 30◦ (D)30◦, 150◦ EXERCISE 4.5. LINEAR PAIR 70

5. [AS1] If the two angles in a linear pair are equal then each of the angles is angle. (A) An acute (B) An obtuse (C)A reflex (D)A right Very Short Answer Type Questions 6 [AS1] Fill in the blanks. is called a linear pair. A pair of adjacent angles whose sum is a 7 [AS2] State true or false. [] (i) A linear pair of angles need not be adjacent. [] (ii) In a linear pair the two adjacent angles are supplementary. [] (iii) The angles of a linear pair form a straight angle. [] (iv) x and y form a linear pair. . (v) x = 180◦ − y [] . EXERCISE 4.5. LINEAR PAIR 71

8 [AS5] Choose the correct answer. (i) Which of the following figures correctly shows a linear pair of angles? (A) (B) (C) (D) EXERCISE 4.5. LINEAR PAIR 72

[AS2] Answer the following questions in one sentence. (ii) Name all pairs of angles that form linear pairs in the figure: (iii) [AS5] Draw two angles forming a linear pair. Short Answer Type Questions 9 [AS2] Saswat’s teacher told him that an angle of 60o and an angle of 120o form a linear pair. Saswat drew them as shown, and his teacher said that they don’t form a linear pair. Who is right? Where did Saswat go wrong? O 10 [AS5] Draw the following pairs of angles as adjacent angles. Check whether they form a linear pair. EXERCISE 4.5. LINEAR PAIR 73

EXERCISE 4.5. LINEAR PAIR 74

EXERCISE 4.6 VERTICALLY OPPOSITE ANGLES 4.6.1 Key Concepts i. Vertically opposite angles: When two lines intersect, the angles that are formed opposite to each other at the point of intersection (vertex) are called vertically opposite angles. ∠1, ∠3 and ∠2, ∠4 form vertically opposite angles. 4.6.2 Additional Questions Objective Questions 1. [AS1] If two straight lines intersect at one point then the number of pairs of vertically opposite angles formed is . (A) 0 (B) 1 (C) 2 (D) 3 2. [AS1] If one of the angles in a pair of vertically opposite angles is 75◦, then the other angle is . (A) 75◦ (B) 15◦ (C) 105◦ (D) 285◦ EXERCISE 4.6. VERTICALLY OPPOSITE ANGLES 75

3. [AS1] In the figure, the vertically opposite angle of ∠AOC is . (A) ∠COD (B) ∠BOC (C) ∠AOD (D) ∠BOD 4. [AS1] In the figure, if ∠BOC = 85◦, then ∠AOD = . (A) 95◦ (B) ◦ (C)275 ◦ 85 (D) None of these 5. [AS1] In the figure, the vertically opposite angle of ∠1 is . (A) ∠4 (B) ∠5 (C) ∠3 (D) ∠6 EXERCISE 4.6. VERTICALLY OPPOSITE ANGLES 76

Short Answer Type Questions 6(i) [AS4] Give two examples of vertically opposite angles that you see in your school or home. (ii) [AS4] Identify the angles formed in the figures given. What is the name of such angles? 7(i) [AS5] Take two straws. Fix them with a pin at their point of intersection, at an angle to one another. What do you observe? State the property. (ii) [AS5] Observe the following figures and name the pairs of angles which are equal in measure. a) b) 77 c) EXERCISE 4.6. VERTICALLY OPPOSITE ANGLES

8(i) [AS5] Draw a pair of intersecting lines and show the vertically opposite angles. (ii) [AS5] Identify vertically opposite angles. 9(i) [AS5] Name all pairs of vertically opposite angles in the figures given. a) b) EXERCISE 4.6. VERTICALLY OPPOSITE ANGLES 78

(ii) [AS5] Name five pairs of vertically opposite angles from the following figure. Long Answer Type Questions 10 [AS2] Find the values of the angles x, y and z in the given figure, without actually measuring them. EXERCISE 4.6. VERTICALLY OPPOSITE ANGLES 79

EXERCISE 4.7 TRANSVERSAL AND PARALLEL LINES 4.7.1 Key Concepts i. A line which intersects two or more lines at distinct points is called a transversal. In the figure (i), t is not a transversal as it doesn’t intersect other two lines at two distinct points. ii. When a transversal intersects a pair of lines, 8 angles are formed. iii. In the figure given, l and m are two lines and ‘t’ is a transversal. The angles formed are ∠1, ∠2, ∠3, ∠4, ∠5, ∠6, ∠7 and ∠8. ∠3, ∠4, ∠5, ∠6 are called interior angles and ∠1, ∠2, ∠7 and ∠8 are called exterior angles. iv. A pair of angles with the following properties are called alternate interior angles. • Both are interior • Non–adjacent • They are on either side of the transversal EXERCISE 4.7. TRANSVERSAL AND PARALLEL LINES 80

v. A pair of angles with the following properties are called alternate exterior angles. • Both are exterior • Non–adjacent • They are on either side of the transversal vi. The following pairs are called interior angles on the same side of the transversal. vii. If alternate angles are equal then the lines are parallel. viii. If two lines are parallel then alternate angles are equal. ix. If corresponding angles are equal then the lines are parallel. x. If two lines are parallel then the corresponding angles are equal. 4.7.2 Additional Questions Objective Questions 1. [AS1] if one of the interior angles on the same side of the transversal intersecting a pair of parallel lines is 55◦ then the other angle is . (A) 35◦ (B) 125◦ (C) 305◦ (D) None of these 2. [AS3] Aline that intersects two or more parallel lines is known as . (A) A transversal (B) A perpendicular (C)A parallel (D) None of these EXERCISE 4.7. TRANSVERSAL AND PARALLEL LINES 81

3. [AS3] If a transversal intersects a pair of lines in two distinct points such that a pair of corresponding angles are equal then the lines are . (A) Perpendicular (B) Parallel (C) Intersecting (D) None of these 4. [AS3] The pair of interior angles on the same side of a transversal intersecting two parallel lines are . (A) Equal (B) Complementary (C) Supplementary (D) None of these 5. [AS3] The pair of alternate interior angles formed by a transversal intersecting two parallel lines are . (A) Equal (B) Complementary (C) Supplementary (D)None of these Very Short Answer Type Questions 6 [AS3] Fill in the blanks. (i) If two lines intersect each other then the number of common points they have is . (ii) The line which intersects two or more lines at distinct points is called a . [AS3] Answer the following questions in one sentence. (iii) Name the pair of angles in the figure by their property. EXERCISE 4.7. TRANSVERSAL AND PARALLEL LINES 82

(iv) Name the pair of angles in the figure by their property. 7 [AS3] Answer the following questions in one sentence. Name the pair of angles in the figure by their property. Short Answer Type Questions 8(i) [AS1] Name the corresponding angles in the figure: (ii) [AS1] Find the values of x, y and z. EXERCISE 4.7. TRANSVERSAL AND PARALLEL LINES 83

9(i) [AS1] Find the angle ‘c’ in the given figure if a : b = 5 : 4. (ii) [AS1] Find the measure of x in the figure, where AB DE. 10 [AS1] Find the measures of the angles of the parallelogram ABCD in which the exterior angle ∠PAB = 47◦. 11(i) [AS3] Name all the angles formed by the transversal EF on lines AB and CD. EXERCISE 4.7. TRANSVERSAL AND PARALLEL LINES 84

(ii) [AS3] Find the values of x, y and z, if the lines 'p' and 'q' are parallel and 'r' is a transversal. Long Answer Type Questions 12 [AS1] Find the measures of all the angles if AB||CD. 13 [AS1] In the adjacent figure, the lines ‘l’ and ‘m’ are parallel and ‘n’ is a transversal. EXERCISE 4.7. TRANSVERSAL AND PARALLEL LINES 85

Find the angles given by stating the appropriate reasons. (i) I f ∠1 = 75◦ then ∠2 = . (ii) I f ∠3 = 50◦ then ∠7 = . (iii) I f ∠2 = 45◦ then ∠8 = . (iv) I f ∠4 = 90◦ then ∠8 = . (v) I f ∠5 = 115◦ then ∠2 = . 14 [AS2] In the given figure, ‘l’ and ‘m’ are intersected by a transversal ‘n’. Is l || m? EXERCISE 4.7. TRANSVERSAL AND PARALLEL LINES 86

CHAPTER 5 TRIANGLE AND ITS PROPERTIES EXERCISE 5.1 CLASSIFICATION OF TRIANGLES 5.1.1 Key Concepts i. Triangles can be classified based on their sides and angles. ii. Based on sides, triangles are of three types. a. Equilateral triangle: A triangle in which all the three sides are equal is called an equilateral triangle. In ∆ABC, AB = BC = CA, also ∠A = ∠B = ∠C. In an equilateral triangle each angle is equal to 60°. b. Isosceles triangle: A triangle in which two sides are equal is called an isosceles triangle. In ∆PQR, PQ = PR also ∠Q = ∠R. c. Scalene triangle: A triangle in which no two sides are equal is called a scalene triangle. In ∆BAT, BA AT BT also ∠B ∠A ∠T. EXERCISE 5.1. CLASSIFICATION OF TRIANGLES 87

iii. Based on angles, triangles can be classified into three types. a. Acute angled triangle: A triangle in which all the three angles are acute (angle that measures less than 90◦) is called an acute–angled triangle. In ∆T AP, ∠T, ∠A and ∠P are acute angles. b. Obtuse angled triangle: A triangle in which one angle is obtuse (angle that measures greater than 90◦) is called an obtuse angled triangle. In ∆FAN, ∠A is obtuse angle. A triangle cannot have more than one obtuse angle. c. Right angled triangle: A triangle in which one angle is a right angle (90 ) is◦called a right–angled triangle. In ∆COT, ∠O is a right angle (i.e., 90◦). A triangle cannot have more than one right angle. Note: Right angled isosceles triangle: A triangle in which one angle is right angle and has two equal sides is called a right angled isosceles triangle. In ∆POT, PO = OT and ∠O = 90◦ ∠P = ∠T = 45◦ . EXERCISE 5.1. CLASSIFICATION OF TRIANGLES 88

Family of triangles – flow chart 5.1.2 Additional Questions Objective Questions 1. [AS1] If the three sides are different in length, then the triangle is known as triangle. (A) An equilateral (B) An isosceles (C)A scalene (D)A right 2. [AS1] If the three sides of a triangle are of the same length, then the triangle is known as triangle. (A) An equilateral (B) An isosceles (C)A scalene (D)A right EXERCISE 5.1. CLASSIFICATION OF TRIANGLES 89

3. [AS1] If one of the angles of a triangle is obtuse then the triangle is known as triangle. (A) An acute angled (B) An obtuse angled (C)A right–angled (D) None of these 4. [AS1] In a triangle if the angles are in the ratio of 1 : 2 : 3, then the triangle is triangle. (A) An acute angled (B) An obtuse angled (C) A right–angled (D) None of these 5. [AS1] In a triangle if the two angles are 57◦ and 49◦ , then the third angle is . (A) 57◦ (B) 49◦ (C) 106◦ (D) 74◦ Very Short Answer Type Questions 6 [AS3] Answer the following questions in one sentence. (i) What do you mean by collinear points? (ii) What is the minimum number of points required to prove whether the given points are collinear or not? (iii) How many lines can be drawn through 3 non–collinear points? (iv) Name the six elements of a triangle. 7 [AS3] Fill in the blanks. (i) A set of points in a plane which extends indefinitely in both the directions form a . (ii) A triangle is formed with 3 points. EXERCISE 5.1. CLASSIFICATION OF TRIANGLES 90

(iii) Triangles can be classified into types based on sides and angles. (iv) The figure formed by joining three collinear points in a plane is called a . (v) Number of right angles a triangle can have is . 8 [AS5] Answer the following questions in one sentence. Mark three collinear points A, B and C and join them. Do you think that the figure you got by joining the three points is a triangle? Short Answer Type Questions 9 [AS2] Classify these triangles based on the lengths of their sides: 10 [AS2] Classify the following triangles based on their angles: Long Answer Type Questions 11 [AS2] Saritha was asked to draw a triangle. She was given the lengths of the sides of the triangles to be drawn as follows. She found that she can use only one of these sets of measurements to construct a triangle. Which one is it? Give reasons for your answer. Set 1: 6 cm, 9 cm and 20 cm Set 2: 5 cm, 7 cm and 12 cm Set 3: 6 cm, 8cm and 12 cm Set 4: 4 cm, 13 cm and 6 cm EXERCISE 5.1. CLASSIFICATION OF TRIANGLES 91

12 [AS2] The lengths of two sides of a triangle are 6 cm and 9 cm. Write all the possible lengths of the third side. EXERCISE 5.1. CLASSIFICATION OF TRIANGLES 92

EXERCISE 5.2 ALTITUDES OF TRIANGLES 5.2.1 Key Concepts i. Altitude of a triangle a. The line segment drawn from a vertex of a triangle perpendiuclar to its opposite side is called the altitude or height of the triangle. b. An altitude can be drawn from each vertex. So, a triangle has three altitudes. c. Altitudes of a triangle are concurrent. d. The point of concurrence of altitudes of a triangle is called the orthocentre of the triangle. ii. Medians of a triangle a. A line segment joining a vertex of a triangle to the mid–point of its opposite side is called a median. b. A median can be drawn from each vertex. A triangle has three medians. c. The medians of a triangle are concurrent. d. The point of concurrence of medians of a triangle is called the centroid of the triangle. EXERCISE 5.2. ALTITUDES OF TRIANGLES 93

In ∆ABC, D, E and F are mid–points of the sides AB, BC and AC. AE, CD and BF are the medians. G is the point of concurrence of the medians, called the centroid. 5.2.2 Additional Questions Objective Questions 1. [AS1] The number of altitudes that can be drawn in a triangle is . (A) 0 (B) 1 (C) 2 (D) 3 2. [AS1] The line segment joining the mid–point of one side of a triangle to the opposite vertex is called . (A) A median (B) An altitude (C)An angle bisector (D) None of these 3. [AS1] The point of intersection of the medians of a triangle is called . (A) Orthocentre (B) Incentre (C) Centroid (D) Circumcentre 4. [AS1] The sum of any two sides of a triangle is the third side. (A) Greater than (B) Less than (C)Equal to (D) None of these 5. [AS1] The difference between any two sides of a triangle is the third side. (A) Greater than (B) Less than (C)Equal to (D) None of these EXERCISE 5.2. ALTITUDES OF TRIANGLES 94

Short Answer Type Questions 6(i) [AS2] Name the altitudes in the following figures: (ii) [AS5] Draw the altitudes for the following triangles. Long Answer Type Questions 7 [AS2] (i) What is the point of intersection of the medians of a triangle called? (ii) Does a median always lie in the interior of the triangle? (iii) How many medians can we draw for a triangle? How do you represent the intersection of the medians? 8 [AS2] (i) Is it correct to say that an altitude of a triangle always lies in the interior of a triangle? (ii) Can you draw altitudes of a triangle such that they are two of its sides? 9 [AS5] Draw the medians of a triangle and mark their point of intersection. EXERCISE 5.2. ALTITUDES OF TRIANGLES 95

EXERCISE 5.3 PROPERTIES OF TRIANGLES 5.3.1 Key Concepts i. Angle – sum property of a triangle: The sum of the interior angles of a triangle is equal to 180° or two right angles. 5.3.2 Additional Questions Objective Questions 1. [AS1] The sum of the three angles of a triangle is . (A) 0 ◦ (B) A right angle (C)2 right angles (D)A complete angle 2. [AS1] If the two angles of a triangle are 30◦ and 77◦, then the third angle is . (A) 30◦ (B) 77◦ (C) 107◦ (D) 73◦ 3. [AS1] One of the angles of a triangle is 80◦ and the other two angles are equal. Then each of the equal angles is . (A) 100◦ (B) 50◦ (C) 80◦ (D) 40◦ 4. [AS1] In a right angled triangle, one of the other two angles is 37◦. Then the remaining angle is . (A) 53◦ (B) 143◦ (C) 90◦ (D) None of these EXERCISE 5.3. PROPERTIES OF TRIANGLES 96