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. . . 48

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 49

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 50

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 51

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 52

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 . . . 53

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 . . . 54

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 55

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 56

(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 57

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 58

(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 59

Long Answer Type Questions 14 [AS5] Name the ﬁgures drawn. EXERCISE 4.1. INTRODUCTION 60

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 61

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 62

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 63

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 64

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 65

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, 66 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 67

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 68

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 69

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 70

[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 71

EXERCISE 4.5. LINEAR PAIR 72

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 73

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 74

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) 75 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 76

(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 77

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 78

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 79

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 80

(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 81

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 82

(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 83

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 84

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 85

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 86

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 87

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 88

(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 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 89

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 90

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 91

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 92

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 93

5. [AS1] In a right angled triangle, the other two angles are in the ratio 2 : 3. Then the angles in order are . (A) 36◦, 54◦ (B) 54◦, 36◦ (C)45◦, 45◦ (D)None of these Short Answer Type Questions 6(i) [AS1] In ∆ABC, ∠A = 45◦ and ∠B = 30◦ ﬁnd ∠C. (ii) [AS1] One angle of ∆ABC is 70◦ and the other two angles are equal. Find the measure of each of the equal angles. 7 [AS1] In a right angled triangle ABC, one of the angles other than the right angle is 40◦ . Show that the third angle is 50◦. 8 [AS2] Write the proof of the angle–sum property of a triangle. Long Answer Type Questions 9 [AS1] The angles of a triangle are in the ratio of 2 : 3 : 4. Find the angles. 10 [AS1] The angles of a triangle are in the ratio 3 : 5: 10. Find the measure of each angle. EXERCISE 5.3. PROPERTIES OF TRIANGLES 94

EXERCISE 5.4 EXTERIOR ANGLE OF A TRIANGLE 5.4.1 Key Concepts i. Exterior angle of triangle: When one side of a triangle is produced, the angle thus formed is called an exterior angle. In ∆COL , the side OL is produced to D. ∠CLD is an exterior angle. The exterior angle of a triangle is equal to the sum of two interior opposite angles. ∠COL + ∠OCL = ∠CLD 5.4.2 Additional Questions Objective Questions 1. [AS1] In a triangle two of the interior angles are 60◦and 45 ◦. Then the exterior angle at the third vertex is . (A) 75◦ (B) 105◦ (C) 285◦ (D) None of these 2. [AS1] In a triangle ABC, ∠A = ◦ and ∠B = 70◦. Then the exterior angle at C is . 50 (A) 60◦ (B) 120◦ (C) 70◦ (D) 50◦ EXERCISE 5.4. EXTERIOR ANGLE OF A TRIANGLE 95

3. [AS1] The exterior angle at one vertex of a triangle is 150 ◦ and one of the opposite interior angles is 87◦.Then the other angle is . (A) 150◦ (B) 87◦ (C) 237◦ (D) 63◦ 4. [AS1] The exterior angle at one vertex of a triangle is 127◦. Then the interior angle at that vertex is . (A) 127◦ (B) 307◦ (C) 53◦ (D) None of these 5. [AS1] If one of the interior angles of a triangle is 117◦, then the exterior angle at that vertex is . (A) 117◦ (B) 63◦ (C) 297◦ (D) None of these Short Answer Type Questions 6(i) [AS1] Find the value of y in the ﬁgure: (ii) [AS1] In the ﬁgure given, ﬁnd the values of x and y. EXERCISE 5.4. EXTERIOR ANGLE OF A TRIANGLE 96

7 [AS1] One of the exterior angles of a triangle is 110◦and its interior opposite angles are equal to each other. Find the measures of the interior angles of the triangle. 8 [AS2] Prove that an exterior angle is equal to the sum of the interior opposite angles of a triangle. Long Answer Type Questions 9 [AS1] (i) One of the exterior angles of a triangle is 120◦and the interior opposite angles are in the ratio 2 : 3. Find the angles of the triangle. (ii) Find the value of angle 'x' in the ﬁgure. EXERCISE 5.4. EXTERIOR ANGLE OF A TRIANGLE 97

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