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# 202110720-PERFORM-STUDENT-WORKBOOK-MATHEMATICS-G07-FY_Optimized

## Description: 202110720-PERFORM-STUDENT-WORKBOOK-MATHEMATICS-G07-FY_Optimized

practice workbook Mathematics Grade 7 Name: Roll No: Section: School Name:

This practice book is designed to support you in your journey of learning Mathematics for class 7. The contents and topics of this book are entirely in alignment with the NCERT syllabus. For each chapter, a concept map, expected objectives and practice sheets are made available. Questions in practice sheets address different skill buckets and different question types, practicing these sheets will help you gain mastery over the lesson. The practice sheets can be solved with the teacher’s assistance. There is a self-evaluation sheet at the end of every lesson, this will help you in assessing your learning gap.

TABLE OF CONTENT • Assessment Pattern: 40 Marks • Assessment Pattern: 80 Marks • Syllabus & Timeline for Assessment Page 1: Chapter 1: Integers Page 10: Chapter 2: Fractions and Decimals Page 19: Chapter 3: Data Handling Page 30: Chapter 4: Simple Equations Page 36: Chapter 5: Lines and Angles Page 46: Chapter 6: The Triangle and Its Properties Page 57: Chapter 7: Congruence of Triangles Page 65: Chapter 8: Comparing Quantities Page 73: Chapter 9: Rational Numbers Page 80: Chapter 10: Practical Geometry Page 87: Chapter 11: Perimeter and Area Page 97: Chapter 12: Algebraic Expressions Page 105: Chapter 13: Exponents and Powers Page 112: Chapter 14: Symmetry Page 122: Chapter 15: Visualising Solid Shapes

AASSSSEESSSSMMEENNTT PPAATTTTEERRNN MMaarrkkss::4400 GrGadraed7e/ 7M/aMthasths Max Internal PAPER: BEGINNER PAPER: PROFICIENT Mark Option Q.No Skill Level Difficulty Level Skill Level Difficulty Level Easy Medium Difficult Easy Medium Difficult Section A (Question Type: MCQ) ···· · ·· 11 Remembering · Remembering · · Remembering · 21 Remembering · Understanding · · ·· ·· 31 Understanding · Understanding · Analysing · 41 Understanding · · Remembering · ·· 51 Analysing Understanding Applying ·· Section B (Question Type: VSA) ····· Understanding ·· · Analysing 61 Remembering · · Applying ·7 1 Understanding · Remembering Applying Understanding 81 91 Understanding · Remembering ·Section C (Question Type: SA) Analysing 10 1 Analysing Understanding ·11 2 Applying Remembering Understanding 12 2 Remembering Remembering Remembering ·13 2 Understanding Remembering 14 3 15 3 Analysing 16 3 Understanding ·Section D (Question Type: LA) 17 3 Remembering ·18 4 Understanding Remembering 19 4 20 4 Remembering Beginner Paper: (Easy: 50%, Medium: 40%, Diﬀicult: 10%) Proﬁcient Paper: (Easy: 40%, Medium: 40%, Diﬀicult: 20%) Easy Question: Remembering questions directly from the text or from the given exercises. (Mostly from content of book or end of chapter exercise). Medium Diﬀiculty Question: In-depth understanding of questions, not necessarily from the text. (Slightly modiﬁed concepts or end of chapter questions). Diﬀicult Question: Question involving creativity like story writing, analysis question like character analysis, justiﬁcation of title or extracts (mostly requires creative and thinking skills).

ASSAESSSSEMSSEMNETNPTAPTATTETRENRN MMaarrkkss::4800 GradGer7ad/ Een7gl/isMhaths Max Internal PAPER: BEGINNER PAPER: PROFICIENT Mark Option Q.No Skill Level Difficulty Level Skill Level Difficulty Level Easy Medium Difficult Easy Medium Difficult Section A (Question Type: MCQ) 11 Applying • Applying • 21 Remembering • Remembering • 31 Analysing • Analysing • 41 Remembering • Remembering • 51 Analysing • Analysing • 61 Applying • Applying • 71 Remembering • Remembering • 81 Applying • Applying • 91 Understanding • Understanding • 10 1 Applying • Applying • Section B (Question Type: VSA) 11 1 • Understanding • Understanding • 12 1 • Applying • Applying • 13 1 • Understanding • Understanding • 14 1 Understanding • Understanding • 15 1 Remembering • Remembering • 16 1 Analysing • Analysing • 17 1 Remembering • Remembering • 18 1 • Understanding • Understanding • 19 1 Understanding • Understanding • 20 1 Understanding • Understanding • Section C (Question Type: SA) 21 2 Analysing • Analysing • 22 2 • Understanding • Understanding • 23 2 • Understanding • Understanding • 24 2 Applying • Applying • 25 2 Remembering • Remembering • 26 2 Applying • Applying • Section D (Question Type: SA) 27 3 • Analysing • Analysing • 28 3 • Remembering • Remembering • 29 3 • Applying • Applying • 30 3 Remembering • Remembering • 31 3 • Remembering • Remembering • 32 3 Remembering • Remembering • 33 3 Understanding • Understanding • 34 3 • Remembering • Remembering • Section E (Question Type: LA) 35 4 Understanding • Understanding • 36 4 Understanding • Understanding • 37 4 Remembering • Remembering • 38 4 • Remembering • Remembering • 39 4 Understanding • Understanding • 40 4 • Analysing • Analysing •

SYLLABUS FOR ASSESSMENT Grade 7 / Maths CHAPTERS PT-1 TE-1 PT-2 TE-2 Chapter 1: Integers ✓ ✓ Chapter 2: Fractions and Decimals ✓ ✓ ✓ Chapter 3: Data handling ✓ ✓ ✓ Chapter 4: Simple equations ✓ ✓ Chapter 5: Lines and Angles ✓ ✓ Chapter 6: The Triangle and its properties ✓ ✓ Chapter 7: Congruence of Triangles ✓ Chapter 8: Comparing quantities ✓ Chapter 9: Rational numbers ✓ Chapter 10: Practical Geometry ✓ Chapter 11: Perimeter and Area ✓ Chapter 12: Algebraic expressions ✓ Chapter 13: Exponents and Powers ✓ Chapter 14: Symmetry Chapter 15: Visualising solid Shapes Periodic Test-1 Assessment Timeline Term 1 Exam 1st July to 31st July Periodic Test-2 23rd September to 21st October Term 2 Exam 16th December to 13th January 17th February to 9st March

LESSON WISE PRACTICE (This section has a set of practice questions grouped into different sheets based on different concepts. By solving these questions you will strengthen your subject knowledge. A self-evaluation sheet is provided at the end of every lesson.)

1. Integers Learning Outcome By the end of this chapter, a student will be able to: • Use the commutative, associative and • Add and subtract positive and negative integers. distributive property. • Learn the properties of addition and subtraction of • Find the additive identity of the given integer. integers. • Find the multiplicative inverse of integers. • Multiply the positive and negative integers. • Find the product of positive and negative integers. Concept Map Commutative Distributive Associative Properties Closure Addition of Subtraction integers of integers Integers Division Multiplication Properties Closure Distributive Commutative Associative 1

1. Integers Key Points • integers, i.e., 1. (a + b) + c = a + (b + c) Numbers 2. (a × b) × c = a × (b × c) We use numbers to count anything. So, what are the for any three integers a, b and c. various types of numbers? 1.  Natural numbers • Existence of identity Natural numbers are counting numbers but do not 1. Zero (0) is the additive identity for integers, i.e., include zero. This is because you cannot count zero. So a + 0�= 0 + a = a for any integer a. numbers 1, 2, 3, 4, 5, 6....... etc are all natural numbers. 2. 1 is multiplicative identity for integers, i.e., 2.  Whole numbers a�×1� � �=1� �× a�=� a for any integer a. All natural numbers along with zero are called whole numbers. For example 0, 1, 2, 3, 4, 5, 6.......etc are all • Integers show distributive property of whole numbers. multiplication over addition, Note:- These types of numbers do not include fractions. From the definition of natural numbers we a × (b + c) = a × b + a × c for any three integers a, b can conclude that every natural or counting number is a whole number. and c. Integers • Product of a positive integer and a negative integer 3. Integers Integers include all natural numbers, zero and is a negative integer, a × (−b) = − ab where a and b negative numbers for example are w-4e, -h3a, v-2e,, -1, 0, 1, 2, 3.......etc are all integers. So now positive integers. • Product of two negative integers is a positive a. Positive integers:- 1,2,3..... b. Negative integers:- -1, -2, -3...... integer, (−a)× (−b) = ab where a and b are positive c. 0 (zero):- which is an integer that is neither integers. negative nor positive. Note:- Integers like whole numbers do not include • Product of even number of negative integers is fractions for example 3.5, ½ etc. positive, whereas the product of odd number of If the number has no sign attached to it as prefix then negative integers is negative. it means that it is a positive number. For example number 3 is really number +3. • When a positive integer is divided by a negative integer or vice-versa and the quotient obtained is Important Points an integer then it is a negative integer. On the number line when we 4 = −2 −2 a. Add a positive integer, we move to the right. b. Add a negative integer we move to the left. −4 = −2 c. Subtract a positive integer we move to the left. 2 d. Subtract a negative integer, we move to the right. When a negative integer is divided by another Properties of integers negative integer then it gives a positive integer, i.e., • Integers are closed under addition, subtraction and −4 = −2 multiplication, which means that sum of integers will also give integers. 2 • Addition and multiplication are commutative for For any integer a,   a ÷1 = a� and a ÷ 0 is not defined. integers, i.e., 1. a + b = b + a 2. a × b = b × a For any two integers a and b. • Addition and multiplication are associative for 2

1. Integers Work Plan COVERAGE DETAILS PRACTICE SHEET CONCEPT COVERAGE Pre-requisite knowledge * P roperties of integers, representation of Number system numbers on number line, rules on signs for addition and subtraction of integers Integers- * Properties of addition and subtraction of PS – 1 Addition and subtraction integers • Closure property under addition • Closure under subtraction Properties * Properties of integers • Commutative property • Additive property Integers • Multiplication of integers PS – 2 Multiplication and Division • Multiplication of positive and negative PS – 3 integer. • Multiplication of two negative integers • Product of three or more negative integers * Properties of multiplication of integers • Closure under multiplication • Commutative Property • Associative Property • Distributive Property * Properties of division of Integers Worksheet for “Integers” PS-4 Evaluation with Self Check or Peer ---- Self Evaluation Sheet Check* 3

PRACTICE SHEET - 1 (PS-1) 1. Temperature in Patna is 20ᵒ C more than 0ᵒ C, Jaipur is 24ᵒ C more than 0ᵒ C and that in Chandigarh is 3ᵒC below 0ᵒ C. Represent the temperature in the form of integers and arrange the cities in descending order of their temperature. 2. Temperature in a city was = 3ᵒ C on a particular Sunday, it dropped down by = 2ᵒ C on the next day. Find the temperature on Wednesday if it increased by 2ᵒ C each on the subsequent days. 3. A man walks 200m north and from there walks 350m south along the same line. If north is measured as positive, how far is the person from starting point? 4. Which is greater? (i) (−8) + (−4) or (−8) + (4) (ii) 12 + (−3) or −12 + 3 5. Write down a pair of integers whose: (i) sum is -5 (ii) difference is -8 (iii) difference is 4 6. Fill in the blanks: (i) (−8) + (−3) = ______ (ii) (−12) − (−5) = ____+ 5 (iii) (−45) + __ = (−45) (iv) (−13) + (−3)� +(�_ )� =(−13) + (−3) + (−4) (v) (−6) + (9) +�(−3)]� =[ − 6 + 9 +� ______ 4

PRACTICE SHEET - 2 (PS-2) 1. Find the following products. (i) ( 2) × ( −3) (ii) (−3)× (−5)× 8 (iii) (−7)× 4× (−11) (iv) (−5)× (−12)× (−7) (v) (12)× (−6)× (−10) (vi) (−3)× (−6)× (−12)× (10) 2. Verify the following: (i) 24× 8 + (−5)]=[ 24× 8 ]+[ 24× −5 (ii) (−21)× (−3) + (−6) = − 21× −3 + − 21× −6 (iii) (−13)× (−7) − (−5)]=[ − 13× −7 ] − [ − 13× −5 3. Find the product using suitable Properties: (i) 35× (−26) + (−26)× 25 (ii) 425× (−30) + (−425)× 20 (iii) ( −36 ) × 103 (iv) 7 × 47 (v) (-30)× (-2)×8× (-5) 4. Divide each of the following: (i) 35 ÷ (−5) (ii) (−120) ÷ (−15) (iii) (−64) ÷ 3 + (−1) (iv) (−80) ÷ (−5) − (−1) (v) (−12) + (−3) ÷ (−5) 5. State true or false: (i) Product of a positive integer with a negative integer is always positive. (ii) Positive integer divided by a negative integer always gives a positive integer as result. (iii) In a sum of positive integer and negative integer, the result is the sign of greater number. (iv) Difference of two negative integers is always negative. (v) Product of four negative integers is positive. 6. In a class test 2 marks are given for each correct answer and -1 deducted for each wrong answer. If there are total 10 questions in the test: (i) Find the total marks obtained by Rahul if he got 6 answers correct. (ii) If Ritu scores 6 marks more than Rahul, How many questions did she answer. 7. An elevator descends at the rate of 10m/sec. If it starts descending from 100m above the ground, find the time taken to reach -1200m. 5

PRACTICE SHEET - 3 (PS-3) 1. Find the product by suitable rearrangement: 25× 2041× (−4) 2. Difference of an integer and −20� is −35 . Find the number. { }3. Evaluate 10 − (10 − (−10 − 10) −10) . 4. Add the product of −7 and −10 to the quotient of −144 and −12. 5. Sum of three integers is −60 . If two integers are −30 and 10 . Find the third integer. 6. Subtract the sum of −105 and 210 from the sum of −95 and −110. 7. From the product of −10 and 13, add the sum of −15 and 32 . 8. Find the product of −13 and the difference of −35 and 47 . 6

PRACTICE SHEET - 4 (PS-4) I. Choose the correct option: 1. How many even numbers are there between 540 and 570? (A) 17 (B) 16 (C) 15 (D) 14 2. What is the sum of successor and the predecessor of ‘n’? (A) n (B) 2n (C) n2 (D) 2 + n 3. What is the product of smallest 2-digit Integer and smallest 3-digit Integer? (A) mallest 4 digit Integer (B) Smallest 3 digit Integer (C) Smallest 5 digit Integer (D) Largest 3 digit Integer 4. A number in which sum of all its factors is equal to twice the number is called (A) Prime Number (B) Composite Number (C) Perfect number (D) Twin primes 5. 57 + (-29) – (-70) _______ 94 – (-1) – (75) (A) < (B) > (C) = (D) not equal 6. The smallest prime number is a/an ________. (A) Composite Number (B) Odd Number (C) Even Number (D) Negative Number 7. The value of 397 × 99 = (A) 39303 (B) 39313 (C) 39323 (D) 39307 8. Solve 98 × (-2) (A) 196 (B) 198 (C) -196 (D) -198 9. Statement A: The predecessor of smallest positive integer is ‘0’. Statement B: The successor of largest negative integer is ‘0’. (A) Both A & B are False (B) Both A & B are True (C) A is true, B is false (D) A is false, B is true 10. Statement A: The product of two same signs (+, -) is positive. Statement B: The product of two same different (+, -) is positive. (A) Both A & B are False (B) Both A & B are True (C) A is true, B is false (D) A is false, B is true II. Short Answer Questions: 1. Show that (-57) × (100 + 3) = [(-57) × 100] + [(-57) × 3] 2. An atom changes to charged particle called ion if it loses or gains electrons. The charge on an ion is the charge of electrons + charge on protons. Write the missing information in the table. Name of ion Proton charge Electron charge Ion charge Hydroid ion +9 ___________ -1 Sodium ion _________ -10 +1 Aluminum ion +13 -10 _________ Oxide ion +8 -10 _________ 3. Simplify the following. (i) (-400) + 100 + (-2) + (-1) + 20 (ii) (-75) + 25 + 27 + (-9) - (-7) 7

PRACTICE SHEET - 4 (PS-4) III. Long Answer Questions: 1. You are at an elevation 380m above sea level, as you start a motor ride. During the ride, your elevation changes by the following meters: 540m, -268m, 116m, -152m, 490m, -844m, and 94m. What is your elevation relative to the sea level at the end of the ride? 2. The temperature was 50ºC in the morning, it increased by 6ºC at lunch time but decreased by 2ºC in the evening, and the temperature decreased by 10ºC in the next morning. What would be the temperature in the next morning? 8

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. Evaluate: 4. A shopkeeper sells a pen at a profit of Rs. 5 and a pencil at a loss of Rs 3. Find his profit or loss for a (i). (−16) + (−5)  (ii). (−69) + (−18)  (2 Marks) month if he sells 150 pens and 100 pencils in that month. (3 Marks) 2. Simplify: (4 Marks) (i) (−35) + (−14)]÷[(10) + (−3) (ii) (−3)× (−12)× (7)× (−11)  5. Verify −68� �× (−36) − (−16) . (3 Marks) 3. Simplify using suitable property (3 Marks) 74× (−15) + (−74)× (35) 9

2. Fractions and Decimals Learning Outcome • Divide the fraction by whole number and fraction. • Solveproblemsbasedonadditionandsubtraction By the end of this chapter, a student will be able to: • Differentiate the types of fractions. of decimals. • Add and subtract fractions. • Multiply and divide decimals by whole number, • Multiply the fraction by a whole number and multiples of 10, decimals. multiply a fraction by another fraction or whole number. Concept Map Types of Operations Addition Fractions on Fractions Subtraction Fractions Multiplication Fractions and Decimals Division Decimals Operations on Decimals Addition Subtraction Multiplication Division 10

2. Fractions and Decimals Key Points Division of Fractions and Decimals The three types of fractions are: Division of Fractions • Proper Fraction – Fractions that represents a part • When a whole number is divided by a fraction, the of a whole. Examples, 1 , 4 whole number is multiplied with the reciprocal of 37 the fraction. Example: 4 ÷ 3 = 4× 8 = 32 • Improper Fraction – Fraction where the numerator 8 33 is greater or equal to the denominator. Example, 7,5 • When a fraction is divided by a whole number, the 43 fraction is multiplied by the reciprocal of the whole number. • Mixed Fraction – Mixed fraction is the combination Example: 1 ÷ 6 = 1 × 1 = 1 of a whole number and a proper fraction. Example, 3 3 6 18 23, 54 • When we divide one fraction by another, we 45 multiply the first fraction with the reciprocal of the second fraction. Multiplication of Fractions and Decimals Example: 2 ÷ 3 = 2 × 7 = 14 5 7 5 3 15 Multiplication of Fraction • The numerators and denominators of two Division of Decimals fractions are multiplied separately and the • When a decimal number is divided by a whole product is written as product of numerators . For number, the whole numbers are divided first and later the decimal point is divided. product of denominators For example, 6.42 √ 3.2 example, 4 × 7 = 4× 7 = 28 • When a decimal number is divided by 10, 100 and 5 3 5× 3 15 100, shift the digits in the decimal number to the left by as many places as there are zeros over 1, to • The product of two proper fractions is lesser than obtain the quotient. each of the fractions multiplied. For example, 45.9 ÷10 = 4.59 • The product of two improper fractions is greater 45.9 ÷100 = 0.459 than the two fractions multiplied. 45.9 ÷1000 = 0.0459 • The product obtained by multiplying a proper and • When a decimal number is divided by another an improper fraction is lesser than the multiplied decimal number, shift the decimal point to the right improper fraction but greater than the multiplied by the equal number of places in both, to convert proper fraction. the divisor to a whole number. Then, divide. For example, 4.2 ÷ 0.7 = 42 ÷ 7 = 6 Multiplication of Decimals • First, multiply the two decimals numbers as whole numbers. • Count the number of digits to the right of the decimal point in both the decimal numbers. • Add the number of digits counted. • Put the decimal point in the product by counting the digits from its rightmost place. • The count should be the sum obtained earlier. Example: 0.30× 5 = 0.15 While multiplying a decimal number with 10, 100 or 1000, shift the decimal point in the number to the right by as many places as there are zeros over 1. For example, 0.56×10 = 5.6 0.56×100 = 56 0.56×1000 = 560 11

2. Fractions and Decimals Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEETS PS – 1 Pre-requisite knowledge, • Making denominator common, LCM of two LCM, Fractions, decimals numbers, division PS – 2 PS – 3 Fractions, operations on • Types of fractions, Addition of fractions, PS – 4 fractions subtraction of fractions PS – 5 Operations on Fractions • Multiplication of fractions by PS – 6 (i) whole number PS - 7 (ii) fractions Self-Evaluation Sheet • Division of fractions by (i) Whole numbers (ii) Fractions Decimals, Operations on • Place value of decimals, Addition of decimals decimals, subtraction of decimals Operation on decimals • Multiplication of decimals by (i) whole number (ii) decimals (iii) Multiples of 10 Operations on decimals • Division of decimals by (i) Whole number (ii) decimals (iii) multiples of 10 Worksheet for “Fractions and Decimals” Evaluation with Self Check or ---- Peer Check* 12

PRACTICE SHEET - 1 (PS-1) 1. Solve: (i) 1 + 5 36 (ii) 4 − 5 11 6 (iii) 5 − 3 7 (iv) 3 1 + 4 23 (v) 4 13 − 6 2 5 2. Arrange the following in ascending order. (i) 3 , 3 , 5   (ii) 2 , 3 , 7 824 5 6 10 3. Length of a rectangle is 6 3 m breadth is 4 1 m. Find the perimeter of the rectangle. 42 4. Salman reads three-fifth of 75 pages of his lesson. How many more pages does he need to complete the lesson? PRACTICE SHEET - 2 (PS-2) 1. Multiply and express as mixed fraction: (i) 3× 7   (ii) 4 × 7 (iii)  5 × 3   23 6 2 (iv) 2 × 3 1 (v) 8 × 5 2 35 33 2. Multiply and reduce to lowest form (i) 3 × 7 (ii) 5 × 3 2 (iii) 9 × 5 1 59 65 72 (iv) 11 × 12 (v) 5 × 21 35 9 10 3. Find 2 of 3 (i) 24 (ii) 15 (iii) 8. 4. In a tank containing 50 liters water is stored. If 2 of the tank is emptied in 3 days, how much water is left 5 in the tank. 5. Which is greater (i) 2 of 5 or 3 of 7 34 53 (ii) 3 of 2 or 8 of 2 45 35 13

PRACTICE SHEET - 3 (PS-3) 1. Divide: (ii) 6 ÷4 (iii) 3 ÷2 2 (i) 3 ÷5 5 (vi) 75 4÷ 8 4 (v) 4 1 ÷6 5 7 14 (iv) 3 2 ÷4 5 37 36 (vii) 3 1 ÷2 1 55 2. Mohan has 2 1 apple. If he divides it among his 3 friends, what will be each person’s share? 2 3. Ram bought butter which weighs 3 kgs. If he divides it into portions that are 1 kg each. How many 48 portions can he make? PRACTICE SHEET - 4 (PS-4) 1. Express 50cm as metre. 2. Express 75vrupees 54 paise in decimal. 3. Express 540 paise as rupees 4. Express 550 gms in kgs. 5. Write in expanded form: (i) 23.45  (ii) 134.64 (iii) 14.564 6. Write the place value of 4 in each of the following: (i) 14.5  (ii) 5.45  (iii) 2.654  (iv) 41.27 7. Vishnu buys 3kg 200gm sugar, 4kg 350 gm rice and 500 gm dal in a bag. What is the total weight of the bag? 8. How much is 36.5 km less than 50.42 km? 14

PRACTICE SHEET - 5 (PS-5) 1. Multiply: (i) 0.4× 7 (ii) 3.45× 4 (iii) 6.57 × 7 (iv) 131.2× 6 (v) 54.8× 9 (vi) 0.2×1.5 (vii) 2.56×1.5 (viii) 3.56×1.45 (ix) 6.5×1.99 (x) 102.5×10.25 2. Find: (i) 1.5×10 (ii) 3.52×100 (iii) 5.646×10 (iv) 3.65×1000 (v) 2.5×100 (vi) 256.8×100 (vii) 0.025×10 (viii) 99.999×100 (ix) 6.52×1000 (x) 0.005×100 3. Find the area of a rectangle with 56.42 m length and 45.62 m as width. 4. Cost of one litre of diesel is Rs 67.05 . Find the cost of 6 litres of diesel. PRACTICE SHEET - 6 (PS-6) 1. Find: (i) �0.50 ÷ 5 (ii) 6.4 ÷ 4 (iii) 12.3 ÷ 6 (iv) 28.72 ÷ 8 (v) 453.9 ÷ 3 (vi) 8 ÷ 2.5 (vii) 63.75 ÷ 0.15 (viii) 10.25 ÷ 2.5 (ix) 0.6 ÷ 0.03 (x) 27 ÷ 0.3 2. Find: (i) 2.5 ÷10� (ii) 3.65 ÷100 (iii) 14.56 ÷10 (iv) 254.25 ÷1000 (v) 0.52 ÷10 (vi) 132.58 ÷1000 (vii) 568.38 ÷100 (viii) 198.526 ÷100 (ix) 0.25 ÷100 (x) 102.56 ÷100 3. A train covers 320km in 1.6 hours. How much time will it take to cover 600 km? 4. Cost of 10 pens is Rs. 65.85 . Find the cost of 15 such pens. 5. A car can travel 560 kms with 13 litres of petrol. How far can the car travel with 5 litres of petrol? 15

PRACTICE SHEET - 7 (PS-7) I. Choose the correct option. 1. The reciprocal of a proper fraction is: (A) a proper fraction (B) an improper fraction (C) a mixed fraction (D) not a number 2. To multiply a decimal number by 1000, we move the decimal point in the number to the _______ side by _______ places. (A) Right side, 3 places (B) Left side, 3 places (C) Right side, 4 places (D) Left side, 4 places 3. ______ is the only number which is its own reciprocal. (A) 0 (B) p (C) 1 (D) ∝ q 4. The reciprocal of 4 2 is: 3 (A) 3 (B) 3 14 (D) 3 2 42 (C) 3 14 5. 2 of 9 is: 3 (A) 18 (B) 6 (C) (D) 3 × 9 3 2 6. 3÷ =9 7 49 (A) 3 (B) 9 (C) 7 (D) 7 7 7 3 9 7. _______ letters comes 2 of the way among A and J? 5 (A) A (B) B (C) C (D) D 8. Pictorial representation of 3× 2 is: 3 (A) (B) (C) (D) 9. Statement A: All fractions are rational numbers. Statement B: Fractions can be negative numbers. (A) Both A and B is true (B) Both A and B is false (C) A is true B is false (D) A is false B is true 10. Statement A: Terminating decimals are called rational numbers. Statement B: Non-terminating decimals are called irrational numbers. (A) Both A and B are true (B) Both A and B are false (C) A is true, B is false (D) A is false, B is true 16

PRACTICE SHEET - 7 (PS-7) II. Short Answer Questions: 1. One packet of biscuits requires 2 1 cups of flour and 1 2 cups of sugar. Estimate total quantity of both 2 3 ingredients used in 10 such packets of biscuits preparation. 2. Solve that  3 of 5  divided by 7.  4 7  11  1 + 1   4 5  3. Show 1 − 3 × 3  8 5  III. Long Answer Questions: 1. Ganga has bought a carpet of size 4m × 6 2 m . But her room size is 3 1 m ×5 1 m . What fraction of 3 3 3 dimensions should be cut off to fit wall to wall carpet into the room? 2. Simplify (0.2× 0.14) + (0.5× 0.91) ( 0.1) × ( 0.2) 17

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. Find 2 1 + 5 . (1 Mark) 6. Find the cost of 5 books each costing Rs. 25.65 37 and 9 notebooks each costing Rs. 15.25. (3 Marks) 2. Find: 12.65× 3.605 . (1 Mark) 3. Find the average of 3.4, 2.9,12.5, 3.09,15.79. (2 Marks) 7. By selling oranges at Rs. 5 1 per orange, a 4 shopkeeper gets Rs. 630. How many dozens did he sell? (3 Marks) 4. Convert 3.56 m into (i) cm (ii) Km. (2 Marks) 5. Find the sum of 3 and −2 and multiply the sum 47 by 2 . (3 Marks) 3 18

3. Data Handling Learning Outcomes At the end of this chapter, students will be able to: • Read bar graph data and construct bar graphs • Organise given data. from given data. • Interpret given data. • Calculate representatives values viz; mean, mode • Determine probability of various cases. and median of given data. Concept Map Key Points • A bar graph is a representation of numbers using bars of uniform widths. • The collection, recording and presentation of data help us organize our experiences and draw • Double bar graphs help to compare two collections inferences from them. of data at a glance. • Before collecting data, we need to know what we • There are situations in our life, that are certain to would use it for. happen, some that are impossible and some that may or may not happen. The situation that may or • The data that is collected needs to be organized may not happen has a chance of happening. in a proper table, so that it becomes easy to understand and interpret. • Average is a number that represents or shows the central tendency of a group of observations or data. • Arithmetic mean is one of the representative values of data. • Mode is another form of central tendency or repre- sentative value. The mode of a set of observations is the observation that occurs most often. • Median is also a form of representative value. It refers to the value which lies in the middle of the data with half of the observations above it and the other half below it. 19

3. Data Handling Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET • Introduction PS – 1 • Organization of data using tally • Determine arithmetic mean PS – 2 PS – 3 Data Handling • Mode and median of data PS – 4 • Interpret data from bar graph PS – 5 • Construct bar graph Self-Evaluation Sheet • Probability Worksheet for “Data Handling” Evaluation with self- check or ---- Peer check* 20

PRACTICE SHEET - 1 (PS-1) 1. Organise the following data in a tabular form. 10. The following are weights (in kg) of 12 people 45 ,65, 44, 43, 23, 44, 55, 43, 65, 9, 6, 6, 43, 6, 6, 70, 62, 54, 57, 62, 84, 75, 59, 62, 65, 78, 60 9, 45 , 6, 23, 55 a. Find the mean of the weights of the people. (i) Which number is the highest? b. How many people weigh above the mean (ii) Which number is the lowest? weight? (iii) What is the range of the data? c. Find the range of the given data. (iv) Find the arithmetic mean. 2. Organise the following data in a tabular form. 6.38, 7.85, 5.55, 5.55, 9.66, 7.55, 7.55, 4.56, 4.56, 7.58, 7.87, 8.32, 7.85, 6.56, 7.55, 8.32, 9.66, 8.32, 8.88, 9.88 (i) Which number is the highest? (ii) Which number is the lowest? (iii) What is the range of the data? (iv) Find the arithmetic mean. 3. Find the arithmetic mean of all multiples of 11 from 1 to 100. 4. Find the arithmetic mean of all prime numbers from 1 to 30. 5. A student scored the following in 8 semesters. Find the mean score. 76, 79.5, 82, 74, 78, 80, 76, 78.5 6. Find the average of even numbers from 1 to 30. 7. Following table shows the data collected from student’s marks: Student Physics Chemistry Mathematics Anuj 75 65 100 Shankar 85 75 90 Ram 88 88 75 Ramesh 65 55 65 Venki 78 98 99 i. Find the average of every student. ii. Determine the highest scorer. 8. Following is the data collected during 7 consecutive years: 21000, 28000, 25000, 23500, 13500, 15500, 20000. Find the range and the mean during this period. 9. The height of 15 students were measured in cm and are as follows: 135, 150, 140, 128, 151, 122, 133, 144, 151, 137, 140, 143, 123, 129, 150. i. Find the range. ii. How many students have height more than the mean height? iii. How many students have height less than the mean height? 21

PRACTICE SHEET - 2 (PS-2) 1 . Find the mode of the following data. 1, 2, 9, 8, 7, 6, 6, 5, 5, 4, 4, 4, 4, 6, 6, 6, 7, 8, 9, 1, 4, 1, 1, 1, 2, 2, 2, 2, 4, 5, 6, 3, 7, 3, 7, 3, 5, 6, 8, 2, 1, 0, 0, 4, 5, 6, 8, 9, 4, 0, 4, 7, 8, 8, 9, 1, 2, 5, 2, 0 2. A dice was thrown 15 times and the outcomes are as follows. Find the mean, mode and median of the data. 5,3,4,1,2,6,4,2,2,3,1,5,6,1,2 3. Find the mean and median of the first 15 odd numbers. 4. Find the mean, mode and median of the following data. 55, 42, 32, 42, 66, 55, 75, 22, 32, 63, 55, 62, 42, 45, 43 5. Find mean and median of the first 15 even numbers. 6. Find the mean and median of numbers from 1 to 100 which are multiple of 6. 22

PRACTICE SHEET - 3 (PS-3) 1. Observe the given graph and answer theSales 3. Observe the given graph and answer the questions that follow. questions that follow. No. of Newspaper Sales Report a. Which class has won the maximum medals? b. What is the mean medals won by class VII in 600 500 running race? 400 c. What is the mean medals won by class VIII in 300 200 running race? 100 d. Find the mean medals won by class VII and 0 class VIII. Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec e. Find the ratio of medals won by class VII and Months class VIII in long jump. i. What is the total sales in the first quarter? 4. Observe the given graph and answer the ii. Which month had maximum sales and which questions that follow. Scale is 1 cm = 10 month had minimum sales? students iii. Calculate the mean of sales in the first 6 a. Find the total number of boys and girls in months. class V. iv. What is the average of sales in the full year? 2. Observe the given graph which represents the b. Which class section has more girls? circulation of newspaper in a town and answer c. Find the ratio of girls and boys in class V. the questions that follow. Newspaper Circulation 500 400 300 200 100 0 Hindi Tamil English Marathi Sanskrit Languages a. What is the total number of newspaper in circulation? b. Find the excess number of newspapers read in Tamil than those in Marathi. c. Which language is the least in circulation? d. Find the mean of newspaper in circulation. 23

PRACTICE SHEET - 3 (PS-3) 5. Consider the following data collected from a survey. Favourite Skoda Honda Maruti Hyundai Audi cars Men 850 800 600 1000 1240 Women 700 600 400 600 1000 i. Draw a double bar graph choosing an appropriate scale. ii. Find the ratio of Honda cars preferred by women and men. iii. Which is the most preferred car? iv. What is the mean of cars liked by men and women? 6. Construct a bar graph with the following data. Days of Mon Tue Wed Thurs Fri Sat Sun the week Items sold 85 92 55 43 73 91 65 a. Calculate the average sale during the week. b. Find the ratio of the minimum and maximum sale. c. On how many days of the week was the scale above the average sales? 7. Construct a double bar graph for the following data of matches won by India and Australia. India 2010 2012 2014 2016 2018 Australia 85 78 80 60 50 78 80 76 70 50 a. Find the ratio of matches won by India and Australia. b. Which team has won the maximum matches? 24

PRACTICE SHEET - 4 (PS-4) 1. A group of students from a German college are selected for a study. The colour of the eyes of the students are noted as: Colour of the Eyes No. of students Black 11 Brown 5 Blue 7 Green 7 Grey 4 What is the probability that the colour of the student’s eyes chosen randomly, start with the letter ‘g’? 2. What is the probability that a student chosen randomly out of 5 girls and 11 boys will be a boy? 3. Two six-sided dice are rolled. What is the probability that the sum will be 9? 4. Two six-sided dice are rolled. What is the probability that both the dice will have the same number? 5. There is a bag full of 42 balls of which 5 are red, 7 are blue, 10 are yellow and rest are green colour balls. Find the probability, a. That a red ball is picked. b. That a green ball is picked. 6. From a deck of cards, Ramu randomly withdraws a card. a. What is the probability that the card is a King? b. What is the probability that the second card withdrawn is also a King? 7. When two dices are rolled what is the probability that both the numbers that appeared will be odd? 8. When 2 coins are tossed what is the probability that the both shows head? 9. What is the probability that when 2 dice are rolled, the sum of the numbers will be divisible by 4? 10. A card is drawn from a deck of playing cards. Find the probability of a. a red Jack b. a black face card c. a card of diamond d. an ace card 25

PRACTICE SHEET - 5 (PS-5) I. Choose the correct option. 1. The difference between the highest and lowest observations in a data is ________. (A) frequency (B) width (C) range (D) height 2. The probability of each event lies between (D) – 1 and 0 (A) 1 and 2 (B) 0 and 1 (C) – 1 and 1 3. Performing a random experiment is called ______. (A) Trial (B) Event (C) Solution (D) Answer 4. The event Gandhiji’s birthday is on 2ⁿᵈ October is a: (A) Less likely (B) More likely (C) Certain event (D) Impossible event 5. Some integers are marked on the rectangle board. What is the range of these integers. (A) 31 (B) 37 (C) 20 (D) 35 6. The mean of 8, 4, x, 6, 2, 7 is 5, then the value of x is _______. (D) 5 (A) 2 (B) 3 (C) 4 7. Which of the following has the same mean, median and mode? (A) 6, 2, 5, 4, 3, 4, 1 (B) 4, 2, 2, 1, 3, 2, 3 (C) 2, 3, 7, 3, 8, 3, 2 (D) 4, 3, 4, 3, 4, 6, 4 8. Let x, y, z be three observations. The mean and mode of these observations are: (A) x × y × z , 3 (B) x+ y+ z, 3 (C) x× y×z , No mode (D) x+ y+ z, No mode 3 3 3 3 9. Statement A: If a fair coin is tossed, occurrence of head and occurrence of tail are equally likely events. Statement B: P(A)= Number of favourable cases to the event 'A' Number of possible out comes (A) Both A and B are false (B) Both A and B are true (C) A is true B is false (D) A is false B is true 10. Statement A: Mode = 3 median – 2 mean Statement B: Central tendency means mean, median and mode (A) Both A and B are true (B) Both A and B are false (C) A is true, B is false (D) A is false, B is true 26

PRACTICE SHEET - 5 (PS-5) II. Short answer questions. 1. Show that mean of first six multiples of 5 is 11.67. 2. The marks in a subject for 12 students are as follows. 31, 37, 35, 38, 42, 23, 17, 18, 35, 25, 35, 39 solve the following using above data. a) Range b) Mode 3. If three fair coins are tossed together, find the probability of getting: a) At least one tail b) At most one tail III. Long answer questions. 1. A board was spun 1000 times and the frequency of outcomes were recorded as in given table. Outcome Red Orange Purple Yellow Green Frequency 185 195 210 206 204 Find (a) list the possible outcomes that you can see in the spinner (b) compute the probability of each out come (c) Find the ratio of each outcome to the total number of times that the spinner spin. 2. Analyse the data and answer the following. 16, 15, 16, 16, 8, 15, 17 a) Which data value can be put in the data so that the mode remains the same? b) At least how many and which values must be put in to change the mode of 15? c) What is the least number of data values that must be put into change the mode 17? 27

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. Find the arithmetic mean of all multiples of 9 4. Find the mean and median of numbers from 1 to from 1 to 100. (1 Mark) 110 which are multiples of 7. (2 Marks) 2. What is the probability that heads will appear when a coin is tossed? (1 Mark) 5. Construct a double bar graph for the following data of marks obtained by class VI and class VII.  (3 Marks) Class VI English Hindi Math Social Science Class VII 85 78 80 85 92 78 80 76 95 87 3. Find the mode and range of the following data. a. Find the ratio of total marks scored by class 1, 2, 7, 8, 7, 6, 6, 5, 5, 4, 4, 4, 4, 6, 6, 6, 7, 8, 9, 2, 4, VI and VII. 1, 1, 1, 2, 2, 2, 2, 4, 5, 6, 3, 7, 3, 7, 3, 5, 6, 8, 2, 1, 0, 0, 4, 3, 6, 8, 9, 4, 0, 4, 7, 8, 8, 9, 1, 2, 5, 2, 1 b. What is mean marks scored by class VI and VII? c. Find the ratio of marks scored in Math and Science by both classes. (2 Marks) 28

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 6. A card is drawn from a deck of playing cards. 7. Observe the given graph and answer the questions that follow. Scale is 1 cm = 10 Find the probability of (3 Marks) students. (3 Marks) a. a red queen b. a heart face card Boys Girls c. a non-ace card 5 4 No. of Students 3 2 1 0 Cricket Hockey Football Badminton Swimming Favourite Sport a. Which sports is most preferred by boys and girls? b. Find the mean of sports played by boys. c. Find the ratio of girls playing cricket and badminton. 29

4. Simple Equations Learning Outcome • Convert equation in the form of statements. • Solve simple equations by trial and error method. By the end of this chapter, a student will be able to: • Solve simple equations by transposing a number. • Define an equation. • Form equations for the given statements or write statements in the form of equations. Concept Map Key Points • Systematic method of solving an equation:- o Carry out series of identical mathematical op- • An equation is a condition on a variable. A variable erations on the 2 sides of the equation in such takes different numerical vales. a way that on one of the sides we just get the variable. • Expressions are formed by performing operations o The last step is the solution of the equation. like addition, subtraction, multiplication and division on the variables. • Transposing means moving to the other side. Transposition of a number has the same effect as • An equation is a condition on a variable such that adding same number to (or subtracting the same 2 expressions in the variable should have equal number from) both sides of the equation. value. • When you transpose a number from one side to the • The value of the variable for which the equation is other side, you change its sign. satisfied is called the solution of the equation. • An equation remains the same if the LHS and RHS are interchanged. • In case of a balanced equation, if we o Add the same number to both the sides or o Subtract the same number from the sides or o Multiply both the sides by the same number or o Divide both the sides by the same number, o The balance remains undisturbed, i.e., the value of the LHS remains equal to the value of the RHS. 30

4. Simple Equations Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET • Balanced equation PS – 1 o Solution to an equation o Forming an equation Simple Equations • Solving equation by separation of variable PS – 2 • Solving equation by transposing PS – 3 • Construction of equations • Form equations and solve PS – 4 Worksheet for “Simple Equations” PS – 5 Evaluation with Self Check ---- Self Evaluation Sheet or Peer Check* 31

PRACTICE SHEET - 1 (PS-1) 1. Say whether the equation is satisfied or not. (i) x + 5 = 0 where x = 5 (ii) y − 6 = 1 where y = 7 (iii) 6x = 18 where x = 2 (iv) x = 2 where x = −6 3 2. Check the value given in the brackets is a solution to the given equation or not. (i) 4n + 3 = 15 (n = 2) (ii) 6x + 6 = 18 ( x = 2) (iii) 6 p − 12 = 24 ( p = −2) 3. Write equations for the following statements: (i) 13 subtracted from twice a number gives 3. (ii) One-fifth of a number is five less than the number. (iii) A number is seven more than one third of itself. (iv) Six times a number is 10 more than the number (v) If 10 is subtracted from half of a number, the result is 4. (vi) Subtracting 5 from P, the result is 2. (vii) Five times a number increased by 7 is 27. 4. Write the following equations in statement form. (i) n + 5 = 19 (ii) 7n + 5 = 19 (iii) 4 p − 3 = 13 (iv) 5x = 25 (v) m3 = 2 (vi) p +3 = 9 (vii) 2m = 7 3 3 PRACTICE SHEET - 2 (PS-2) 1. Set up an equation in the following cases: (i) Mohan is 3 years older than Sohan. The sum of their ages is 43. (ii) Age of Sohan is four times that of Amith. The difference of their age is 27 years. (iii) S even times a number is 12 less than 13 times the same number. (iv) In a school, the number of girls is 50 more than number of boys. Total number of students is 1070. (v) In a family, the consumption of wheat is 4 times that of rice. The total consumption of the 2 cereals if 80 kg. 2. Give the steps you will use to separate the variable and then solve the equation. (i) x − 3 = 0 (vii) 40t = −20 (ii) y + 6 = 4 (viii) 3n − 15 = 21 (iii) z − 3 = 3 (ix) 5 p − 10 = 4 (iv) y − 4 = −8 (x) 2x + 8 = 12 (v) 2z = 8 (xi) 210p = 8 (vi) a = 1 62 32

PRACTICE SHEET - 3 (PS-3) 1. Solve the following equations. (i) m + 3 = 2 (ii) x − 7 = 5 6 4 (iii) 2 x − 37 = 7 (iv) 32 y = 3 2 2 2 (v) 6 y = −12 (vi) 9x − 14 = 17 22 (vii) x = 7 (viii) 2m + 6 = 12 5 15 (ix) 5x − 6 = 3x − 8 (x) 3 x + 5 = 5x − 125 (xi) 2(2x − 7) + 4(3x + 2) = 6(5x + 9) + 12 46 3 2. Construct 3 equations with y = 5. PRACTICE SHEET - 4 (PS-4) 1. A number exceeds another number by 12. If their sum is 72, find the numbers. 2. Seven times a number is 12 less than 13 times the same number. Find the number. 3. Interest received by Karim is Rs. 30 more than that of Ramesh. If the total interest received by them is 70, find the interest received by Ramesh. 4. One subtracted from 1 of a number gives one. Find the number. 3 5. The perimeter of a rectangle is 40 m. The length of the rectangle is 4 m less than 5 times its breadth. Find the length of the rectangle. 6. Each of equal sides of an isosceles triangle is twice as large as the third side. If the perimeter of the triangle is 30 cm, find the length of each side. 33

PRACTICE SHEET - 5 (PS-5) I. Choose the correct option. 1. A true mathematical statement containing the sign “=” is called an: (A) equation (B) equality (C) inequation (D) open sentence 2. A particular number written in the place of a variable in an open sentence is called: (A) Addition (B) Multiplication (C) Replacement (D) Elimination 3. If a = c ⇒ ad = bc then the process is called: bd (A) cross multiplication (B) division (C) elimination (D) replacement 4. If 4x + 2 = x + 8, then x = (A) 1 (B) 2 (C) 3 (D) 4 5. The value of ‘m’ which changes m(m + 1) = 30 into a true statement is: (A) 6 (B) 5 (C) 4 (D) – 5 6. By replacing the variable by a number, an open sentence is changed into: (A) Mathematical statement (B) Numerical expression (C) Replacement value (D) Equation 7. A number divided by 2 and then increased by 5 is 9 (A) x + 5 = 9 (B) 2x + 5 = 9 (C) 2 + 5x = 9 (D) 5x + 2 = 9 2 8. If , then the value of 5x + 3 is (A) 32 (B) 33 (C) 10.5 (D) 5 9. Statement A: 3 + 14 is a mathematical statement. Statement B: 3x – y + z = 9 is a mathematical statement. (A) Both A and B are true (B) Both A and B are false (C) A is true B is false (D) A is false B is true 10. Assertion A: 3x + 1 = 0 is a linear equation in one variable. Reason R: ax + b = 0 is a linear equation in one variable when a ≠ 0 (A)Both A and R are correct and R is the correct explanation of A. (B) Both A and R are correct and R is not correct explanation of A. (C) A is in correct, R is correct. (D) A is in correct, R is correct. II. Short Answer Questions: 1. Explain the steps to be followed in solving a verbal (word) problem? 2. Vandana is four times as old as her brother Akash at present. After 10 years, she will be twice the age of her brother. Write and simplify the equation. 3. Solve 5x −4 = 3x + 11 2 4 III. Long Answer Questions: 1. The length of a rectangle is 5cm more than its breadth. If the perimeter of the rectangle is 42 cm, then find the area of the rectangle. 2. Simplify 1 x2 y2 + 9x2 y2 − 3 xy 2 = 5xy2 8 4 34

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. Write the equation x = x − 5 in the statement 5. Arjun has three times as much money as 5 Prateek and Shoba has half as much as Arjun form. (1 Mark) and Prateek put together. If Arjun has Rs. 120, then how much money do Prateek and Shoba have? (3 Marks) 2. If p = 2 then find the value of 1 (1 − 3 p) .(1 Mark) 3 6. In ∆ABC, ∠A is half of ∠B and ∠C is thrice of ∠A. Find the values of ∠A, ∠B and ∠C. (3 Marks) 3. After 20 years, Joey will be 5 times as old as he is now. Find his present age. (2 Marks) 4. What is the difference between an equation and an expression? (2 Marks) 7. Check whether the following are satisfying the condition of a simple equation. (3 Marks) (i) 6 x + 4x + 4 3 (ii) 2t + 34 = 42 (iii) 2x2 + 4x + 2 = 0 35

5. Lines and Angles Learning Outcome By the end of this chapter, a student will be able to: • Identify lines, line segments, angles and rays. • Determine the various related angles. • Identify the various pairs of lines. • Determine parallel lines. Concept Map Key Points • A line segment has 2 end points, If we extend the 2 points in either direction endlessly, we get a line. Line segment PQ is represented as PQ P Q o Supplementary angles: When the sum of the measures of the 2 angles is 180°. Each angle is • A line has no end point. said to be the supplement of the other. A line AB is represented as AB Example: AB 145° 35° • A ray has one end point which is its starting point. Ray OP is represented as OP o Adjacent angles are 2 angles that have a common vertex and a common side but no OP common interior points. • Angle is formed when lines or line segments meet. Example: ∠AOB and ∠AOC are adjacent angles. A C R AO B P S o A linear pair is a pair of adjacent angles whose non-common sides are opposite rays. Q BC Example: ∠AOC and ∠AOB are a linear pair. • Related angles: o Complementary angles: When the sum of the measures of 2 angles is 90°. Example: 36

5. Lines and Angles AB 12 OC 34 l o Vertically opposite angles are the angles 56 m opposite to each other when two lines cross. 87 R 1 Interior angles are ∠3, ∠4, ∠5, ∠6 P 4 S Exterior angles are ∠1, ∠2, ∠7, ∠8 Pairs of Corresponding angles are ∠1 & ∠5, ∠2 & 3 2 ∠6, ∠3 & ∠7, ∠4 & ∠8 Pair of alternate interior angles are ∠3 & ∠6, ∠4 & Q ∠5 Pair of alternate exterior angles are ∠1 & ∠8, ∠2 & Example: ∠7 ∠1 & ∠2 are vertically opposite angles Pair of interior angles on the same side of the transversal are ∠3 & ∠5, ∠4 & ∠6 ∠2 & ∠4 are vertically opposite angles. o Transversal of Parallel lines If 2 parallel lines are cut by a transversal, each pair When two lines intersect, the vertically opposite of corresponding angles are equal in measure. angles so formed are equal. If 2 parallel lines are cut by a transversal, each pair In the figure, of alternate interior angles are equal. ∠1 = ∠2 and ∠3 = ∠4 If 2 parallel lines are cut by a transversal, then each pair of interior angles on the same side of • Pairs of lines transversal are supplementary. o Intersecting lines: Two lines are said to be When a transversal cuts two lines such that pairs intersecting when they have one common of corresponding angles are equal, then the lines point. This common point is called point of have to be parallel. intersection. When a transversal cuts 2 lines, such that pairs of alternate interior angles are equal, the lines have o Transversal: A line that intersects 2 or more to be parallel. lines at distinct points is called a transversal. When a transversal cuts 2 lines, such that pairs of interior angles on the same side of transversal are P is a transversal to the lines l and m . supplementary, then the lines have to be parallel. P l m Angles made by a Transversal: 37