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

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

5. Lines and Angles Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET PS – 1 • Related Angles PS – 2 Lines and Angles • Pairs of Lines PS – 3 Self Evaluation Sheet • Parallel Lines Worksheet for “Lines and Angles” Evaluation with Self Check or Peer ---- Check* 38

PRACTICE SHEET - 1 (PS-1) 1. Find the complementary angle of each of the (i) Corresponding angles following angles: (ii) Pair of alternate interior angles (iii) Vertically opposite angles. 65° (iv) Pair of alternate exterior angles 45° (v) Pair of interior angles on the same side (i) (ii) (iii) 15 2. Find the supplementary angle of each of the 26 following angles: 60° 37 30° 48 3. Identify which of the following pairs of angles 7. Find the values of X in each of the figures below are complementary and supplementary angles: if P || Q (i) 65ᵒ, 115ᵒ R (ii) 37ᵒ, 53ᵒ PQ (iii) 10ᵒ, 80ᵒ (iv) 45ᵒ, 135ᵒ P 30° O 4X R XQ 2X A 4. In the figure below: S (ii) (i) Are ∠2 and ∠5 supplementary angles? (i) (ii) Is ∠1 adjacent to ∠5 ? (iii) What is vertically opposite angle to ∠3 ? 8. Find the value of unknown angles (iv) Is ∠2 vertically opposite to ∠4 ? BC 12 9. In the below figure, OB is perpendicular to OA A 5 O3 and ∠BOC = 50° . Find ∠AOD. 4 B C D 50° E OA 5. Find the values of the angles X, Y and Z. D C B 50° X A YO D Z E 10. In the below figure, find a, b, c. Lines l, m and F n are parallel and p and q are parallel. 6. In the adjoining figure, identify 39

PRACTICE SHEET - 1 (PS-1) n m l 4c 3b P O S 6a 120° QR T 40

PRACTICE SHEET - 2 (PS-2) 1. Find the angle, if the complement angle is 65ᵒ. 8. In the below figure, AB || CD find X. 2. Which of the following pairs are complementary 9. In the below figure, find X and all the angles. angles? (i) 60ᵒ, 30ᵒ C D (ii) 15ᵒ, 65ᵒ 3X-10 (iii) 30ᵒ, 45ᵒ (iv) 45ᵒ, 45ᵒ 4X+10 (v) 85ᵒ, 15ᵒ (vi) 15ᵒ, 17ᵒ 2X+9 3. Which of the following pairs are supplementary angles? (i) 90ᵒ, 90ᵒ (ii) 135ᵒ, 45ᵒ (iii) 170ᵒ, 10ᵒ (iv) 90ᵒ, 90ᵒ (v) 156ᵒ, 34ᵒ (vi) 112ᵒ, 22ᵒ 4. Find the value of X. D 4x+20 AO B 10. Find the values a and b if l || m . l 3x-50 ts A BC A 5. The difference of two complementary angles is 20ᵒ. Find the angles. b a 6. In the below figure, FA EB CD . Find ∠PQR . AB 35° Q C R 640 130 0 P 115° C m FE B D 7. In the below figure, PQ and ST intersect at O. If ∠PQR = 90° and x : y = 3 : 2 , find z. S z O P yQ x T R 41

PRACTICE SHEET - 32 (PS-23) I. Choose the correct option. 1. 1° (one degree) = (A) 40' (B) 50' (C) 60' (D) 30' 2. A right angle is divided into 90 equal parts and each part is called: (D) Right Angle (A) Angle (B) Minute (C) A degree (D) Same angle 3. Points are \"coplanar\" if and only if they lie in the ________. (A) Same plane (B) Same line (C) Same point 4. The intercept in the given figure is: (A) AB (B) CD (C) EF (D) MN 5. The condition between A and B, if l || m is: (D) A ≠ B (A) A > B (B) A = B (C) A + B = 180 6. If OC is the bisector of ∠AOB whose measure is 180°, then ∠AOC = (A) 360° (B) 90° (C) 45º (D) 180º 7. ∠1− ∠8 = [from diagram] (A) 180° (B) 60° (C) 120º (D) 0º 8. From above diagram an allied angle to ∠3 is: (A) ∠1 (B) ∠2 (C) ∠6 (D) ∠8 9. Statement (A): The length of the transversal between the parallel lines is called intercept. Statement (B): A straight line in which intersects two or more lines at the same point is called a trans- versal. (A) Both A and B are true (B) Both A and B are false (C) A is true and B is false (D) A is false and B is true. 10. Statement (A): If ∠PQR = 135, then 4 ∠PQR = 175º 3 Statement (B): The bisector of an Angle divides it into two equal angles. (A) Both A and B are true (B) Both A and B are false (C) A is true and B is false (D) A is false and B is true. 42

PRACTICE SHEET - 3 (PS-3) II. Short answer questions. 1. In the following figure, if AB || CD and ∠EMB = 70º , then by applying appropriate properties find ∠MNC . 2. In the following figure OA || PC and OB || PO then simplify ∠x + ∠y . 3. If in the given figure, PQ || BC∠ABC = 80º and ∠BCA = 40º , then show that ∠x + ∠y + ∠z = 180º. III. Long answer questions. 1. In the following figure. p||q and ‘r’ is the transversal intersecting the lines ‘p’ and ‘q’. If ∠A : ∠B = 2 : 3 , then find all the other angles. 2. In the given figure, XY || PQ, simplify ∠x + ∠y . 43

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. If 70ᵒ is divided in the ratio of 2:5, 4. In the figure below, identify the corresponding what is the value of the smaller part? (1 Mark) angles and alternate interior angles. (2 Marks) 14 23 58 67 2. In the below figure, find the value of X. (1 Mark) ST 40° P 3X 4X R Q 3. In the below figure, AD divides ∠BAC in the 5. (a) Prove that when two lines intersect, the ratio 1:3. Find the value of x and ∠ADC . vertically opposite angles are equal. (2 Marks) (b) Find the value of x from the given figure. E (3 Marks) A 1200 AB 60° O x DE 1000 x BD C 44

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 6. In the given figure, OC = OD, ∠OCD = 30ᵒ. Find 7. (a) In the given figure, Prove that LX MY NZ . ∠AOB.  (3 Marks) (b) If k = 50° , find ∠PSY . (3 Marks) A C P O LX k M kS Y D B kR N Z Q B A 45

6. The Triangles and Its Properties Learning Outcome By the end of this chapter, a student will be able to: • Explain the properties of exterior angle of a • Identify the elements of a triangle. triangle. • Classify triangles. • Determine the median and altitude of a triangle. • Identify the different types of triangles. • Determine the exterior of a triangle. • Apply the property of the lengths of sides of a triangle. • Apply the Pythagoras property of triangle. Concept Map Key Points • Classification of triangles • Triangle is a simple closed curve made of 3 line Triangles segments. Based on sides Based on angles • Elements of a triangle: (i) 3 sides - AB , BC , CA Scalene Isosceles Equilateral Acute Obtuse Right angled (ii) 3 angles - ∠BAC, ∠ABC, ∠BCA (iii) 3 vertices – A, B, C o Equilateral triangle: A triangle in which all three sides are equal in length. Each angle A measures 60°. BC • Angle sum property of a triangle: The sum of 3 angles of a triangle is 180°. 46

6. The Triangles and Its Properties A A 600 BDC • The perpendicular line segment from the vertex of 600 600 B C a triangle to its opposite side is called altitude of a triangle. A triangle has 3 altitudes. o Isosceles triangle: A triangle in which any 2 In ∆ABC, AD is the altitude. sides are of the same length. The non-equal side is called its base. Base angles of an A isosceles triangle have equal measure. A BC BD C o Scalene Triangle: A triangle which has three • Exterior angle of a triangle is formed when a side unequal sides. of a triangle is produced. ∠ACD is the exterior angle of ∆ABC formed at o Acute angled triangle: A triangle with 3 acute vertex C. angles. A o Obtuse angled triangle: A triangle with one obtuse angle and 2 acute angles. B CD o Right angled triangle: A triangle in one angle • Exterior angle property of a triangle: An exterior is a right angle. The side opposite to the right angle of a triangle is equal to the sum of its interior angle is called hypotenuse. The other two opposite angles. sides are called legs. In ∆ABC, ∠ACD = ∠1+ ∠2 A A A 1 A 2 450 250 350 Hypotenuse B CD • Property of lengths of sides of a triangle: The sum Legs of lengths of any 2 sides of a triangle is greater than the length of the third side. 650 700 1150 400 550 • Pythagoras property: In a right angled triangle, the BC square on the hypotenuse is equal to the sum of BC BC squares of its legs. (i) Acute angled (ii) Obtuse angled If a triangle is not right angled, then the Pythagoras triangle (iii) Right angled property does not hold good. This property is useful triangle triangle to decide whether a given triangle is right angled or not. • The line segment joining the vertex of a triangle to the midpoint of its opposite side is called median of a triangle. A triangle has 3 medians. In ∆ABC, AD is the median. 47

6. The Triangles and Its Properties Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET PS – 1 • Classification of Triangle • Elements of a Triangle PS – 2 • Exterior angle property PS – 3 PS – 4 Triangles and its Properties • Angle sum property PS - 5 • Exterior angle property Self Evaluation Sheet • Isosceles and Equilateral Triangle • Property of Length of side • Right angled triangle • Pythagoras property Worksheet for “Triangles and Its Properties” Evaluation with Self Check or ---- Peer Check* 48

PRACTICE SHEET - 1 (PS-1) 1. In ∆ABC, D is the midpoint of   AB . 7. Find the values of the unknown exterior angles. (i) CE is __________ (ii) CD is __________ i) ii) iii) x x (iii) Is AE = EB ? 650 450 500 600 500 300 A DE B x 8. Find the values of the unknown interior angle. C i) ii) 650 2. If the exterior angle of a triangle is 130ᵒ and its 110 0 300 interior opposite angles are equal, then find the x 105 0 measure of each interior opposite angle. x 3. In ∆ABC, AD is the bisector of ∠A meeting BC at D. CF ⊥ AB and E is the midpoint of AC. Find the median of the triangle. A E F Median BD C 4. In ∆PQR, if ∠P = 60° and ∠Q = 40° then determine the exterior angle formed by producing QR. Ans: P 600 400 x QR 5. How many altitudes does a triangle have? 6. Draw a triangle which will have its altitude outside the triangle. 49

PRACTICE SHEET - 2 (PS-2) 1. Determine the measure of angle x in the 6. In the figure, find the measure of ∠PON and following cases. ∠NPO . LM i) B ii) P R 700 200 A 1050 x iii) O 45° x ? x 30° xx ? 700 PN CQ 7. In the figure, if QP || RT , find the value of x and y. 2. Find the values of the unknowns x and y . P i) 300 ii) iii) x 600 x 600 xT y y 600 500 x y 600 3. In the figure, determine the value of x. 300 700 Q y A R xE 250 600 C 350 D B 4. In ∠ABC , ∠A = 100° . AD bisects ∠A and AD ⊥ BC . Determine value of ∠B . 5. In ∆ABC, ∠A = 50°, ∠B = 70° and bisector of ∠C meets AB in d. Determine the measure of ∠ADC . A 500 ? D 700 C B 50

PRACTICE SHEET - 3 (PS-3) 1. The sides of a triangle have lengths (in cm) 10, 6.5 and α where α is a whole number. Determine the minimum value that α can take. 2. ∆DEF is a right angled triangle with ∠E = 90° . What type of angles are ∠D and ∠F? 3. In the figure, if PQ = PS , determine the value of x. P x 110 0 S 250 Q R 4. In an isosceles triangle, one angle is 70ᵒ. Determine the other 2 angles. 5. In ∆PQR, if PQ = PQ and ∠Q = 100° , then determine ∠R. 6. In the figure, BC = CA and ∠A = 40° . Determine ∠ACD . A 400 B CD 7. Length of 2 sides of a triangle are 7 cm and 9 cm. Determine the range of length between which the third side can lie. 51

PRACTICE SHEET - 4 (PS-4) ii) 6 cm x x 8 cm 1. Find the unknown length x. i) 15 cm 8 cm 2. In the right angled triangle ABC, if ∠B = 90°, BC = 3 cm and AC = 5 cm, then find the length of side AB. 3. Foot of a ladder is 6 m away from its wall and its top reaches a window 8 m above the ground. (i) Find the length of the ladder. (ii) If the ladder is shifted in such a way that its foot is 8 m away from the wall, to what height does its top reach? 4. 2 poles 10 m and 15 m stand upright on a plane ground. If the distance between the tops if 13 m, then find the distance between their feet. 5. In a right angled triangle, if one of the non-right angles is 30ᵒ more than the other non-right angle, then find the angles of the triangles. 6. Height of a pole is 8 m. Find the length of the rope tied with its top from a point on the ground at a distance of 6 m from its bottom. 7. Find the perimeter of a rectangle whose length is 60 cm and diagonal is 61 cm. 8. Diagonals of a rhombus are 12 cm and 16 cm. Find its perimeter. 52

PRACTICE SHEET - 5 (PS-5) I. Choose the correct option. 1. The number of components or parts of a triangle is: (A) 3 (B) 5 (C) 6 (D) 9 2. A triangle having angles 90º, 45º, 45º is called: (A) acute angled triangle (B) obtuse angled triangle (C) Right angled isosceles tringle (D) scalene triangle 3. An exterior angle in an equilateral triangle is: (A) 60º (B) 80º (C) 100º (D) 120º 4. A triangle divides a plane in which it lies into: (A) five parts (B) two parts (C) three parts (D) four parts 5. The sum of the measures of the angles of a triangle is: (A) 180º (B) 2 right angles (C) a straight angle (D) All the above 6. In an equilateral triangle ABC, if AB = x cm, BC = y cm, CA = z cm, then: (A) x > y > z (B) x < y < z (C) x = y ≠ z (D) x = y = z 7. In a ∆ABC, if ∠B = 90º , AB = x cm, BC = y cm and AC = z cm, then: (A) z2 = x2 + y2 (B) x2 = z2 - y2 (C) y2 = x2 - z2 (D) x2 = y2 +z2 8. In the adjoining figure, ABC is a triangle in which ∠A = 80º and ∠B = ∠C then ∠B and ∠C are respectively (A) 30º, 30º (B) 60º, 60º (C) 40º, 40º (D) 50º, 50º 9. Statement (A): All polygons are triangle. Statement (B): The sides of a triangle represented by using upper case letters. (A) Both A and B are true (B) Both A and B are flase (C) A is true and B is false (D) A is false and B is true 10. Statement (A): Every equilateral triangle is an isosceles triangle. Statement (B): A triangle having 3 obtuse angles is called as obtuse angle triangle. (A) Both A and B are true (B) Both A and B are false (C) A is true and B is false (D) A is false and B is true II. Short answer questions. 1. In ∆ABC and bisector of LC meets AB in D show than ∠D = 110º . 2. Show the following are the measures of the sides of triangle AB = 3 cm, BC = 4 cm, CA = 5 cm 3. Apply Isosceles triangle angle property and find xº in following diagram. 53

PRACTICE SHEET - 5 (PS-5) III. Long answer questions. 1. In a ∆ABC, AB = AC and with data as shown in figure. Find x, y. 2. In ∆ABC ∠A - ∠B = 33ᵒ and ∠B - ∠C = 18ᵒ. Check all the angles of ∆ABC are acute angles. 54

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. Find the value of the angle x . (1 Mark) 3. In the figure, find the measure of ∠COD and 1300 ∠CDO .  (2 Marks) x 650 C A 450 150 O 650 x B D 2. Define median of a triangle. (1 Mark) 4. In the figure AB || CD . Find the value of x and y.  (2 Marks) AC40° y 60° xD B 55

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 5. Diagonals of a rhombus are 6 cm and 8 cm. Find 7. Jayanti takes shortest route to the home by its perimeter. (3 Marks) walking diagonally across a rectangular park. The park measures 60 m � 80 m. How much shorter is the route across the park then the route around its two adjacent angles edges? (3 Marks) 6. In the figure, if AB = BC , determine the value of x.  (3 Marks) DCA 300 1200 x B 56

7. Congruence of Triangles Learning Outcomes At the end of this chapter, students will be able to: • Identify congruent triangles. • Understand congruency. • Apply SAS, SSS, ASA & RHS congruency criteria to • Identify congruent plane figures, line segments and triangles. angles. Concept Map Key Points o RHS Congruence Criterion • SSS Congruence of 2 triangles: Under a given • Congruent objects are exact copies of one another. The relation between 2 objects being congruent is correspondence, 2 triangles are congruent if 3 sides called congruence. of one triangle are equal to the 3 corresponding sides of other triangle. • Congruence of two plane figures can be examined • SAS Congruence of 2 triangles: Two triangles are by the method of superposition. Two plane figures congruent if 2 sides and the angle included between P1 and P2 are congruent if trace-copy of P1 fits them in one triangle are equal to the corresponding exactly on that of P2 . sides and the angle included between them of the other triangle. • Congruence of P1 and P2 is written as P1 ≅ P2 • ASA Congruence of 2 Triangles: Two triangles are • Two Line segments   AB and   CD are congruent if congruent if 2 angles and the side included between them in one of the triangles are equal to the they have equal lengths. This is written as   AB ≅ CD corresponding angles and the side include between or   AB = CD. them in the other triangle. • Two angles, ∠ABC and ∠PQR are congruent if • RHS Congruence of 2 right angled triangles: their measures are equal. This is written as Two right angled triangles are congruent if the ∠ABC ≅ ∠PQR or as m∠ABC = m∠PQR or simply hypotenuse and the base of one of the triangles are as ∠ABC = ∠PQR. • Congruence of Triangles: Three criteria applicable are o SSS Congruence Criterion o SAS Congruence Criterion o ASA Congruence Criterion 57

7. Congruence of Triangles • equal to the hypotenuse and corresponding base of the other triangle. • There is no AAA congruence of 2 triangles. Two triangles with equal corresponding angles need not be congruent. In such a correspondence one of them can be enlarged copy of another. Two figures are congruent if and only if they are exact copies of each other. Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Congruence of Triangles • Define Congruence PS – 01 • Congruent Plane figures • Congruent Line segments PS – 02 • Congruent Angles Self-evaluation Sheet • Explain criterion of congruence - SSS, SAS, ASA, RHS • Numerical on SSS, SAS, ASA, RHS Worksheet for “Congruence of Triangles” Evaluation with self- check or ---- Peer check* 58

PRACTICE SHEET - 1 (PS-1) 1. If ∆PQR ≅ ∆LMN under the correspondence PQR ↔ LMN , write all the corresponding parts of the triangles. 2. In the following figure, by which criterion of congruency are the triangles congruent? DP 80° 70° 30° 70° i) E FQ R ii) 3. In the triangle ABC and PQR, ∠B = 90°, AC = 8 cm, BC = 3 cm, ∠P = 90°, PR = 3 cm and QR = 8 cm. By which criterion of congruence are the two triangles congruent? 4. In triangles DEF and PQR, ∠E = 80°, ∠F = 30°, EF = 5 cm, ∠P = 80°, PR = 5 cm, ∠R = 30° . By which criterion of congruence are the 2 triangles congruent? 5. In the given figure, by which criterion of congruence are the triangles congruent? D AC B 6. If AB = 5 cm, QR = 7 cm, then find the value of AC (in cm) if the perimeter of ∅ABC is 18 cm and ∆ABC ≅ ∆PQR. A BC PQ R 59

PRACTICE SHEET - 1 (PS-1) 7. In the figure, if ∠A = 50° and ∠Q = 60°, then find the value of ∠B (in degrees). 8. In the figure, ∆BOC ≅ ∆AOD and ∠DOA = 30°, then what is the measure of ∠BCO (in degrees)? 9. If ∆DEF ≅ ∆BCA which part of ∅BCA corresponds to ∠E ? 10. Line AB of 4 cm is drawn. Another line CD which is congruent & perpendicular to AB is drawn. What is the length of CD? 60

PRACTICE SHEET - 2 (PS-2) I. Choose the correct option. (A) RHS 1. If measure of corresponding side and areas are (B) ASA equal, then those figures are ______. (C) SSS (A) Similar (D) SAS (B) Congruent (C) similar and congruent 7. If the exterior angle of a triangle is 130º and its (D) None of the above interior opposite angles are equal, then mea- 2. The congruency of two figures is represented sure of each interior opposite angle is: by the symbol______. (A) ~ (B) ≅ (C) −−~=~ (D) 3. The point of concurrence of the medians of a (A) 45º triangle is _____. (B) 55º (A) Centroid (C) 65º (B) Ortho centre (D) 75º (C) In centre (D) Excentre 8. By applying bisector, the point of concur- 4. In the adjacent figure, line 'l' is called: rence of internal bisector of one angle and the external bisectors of other two angle is (A) Median _______. (B) Altitude (A) In centre (C) Perpendicular bisector (B) Excentre (D) Angular bisector (C) Circum centre (D) Ortho centre 5. In ∅PQR is congruent to ∅STU , then the 9. Statement (A): In the figure ∆ABD =~ ∆ABC length of TU is: Statement (B): If the hypotenuse and side of one triangle are equal to the corresponding hypotenuse and side of another right angle triangle is RHS (A) Both A and B are true (A) 5 cm (B) Both A and B are false (B) 6 cm (C) A is false, B is true (C) 7 cm (D)A is true, B is false (D) cannot be determined 10. Statement (A): A line segment which joins a 6. By which of the following criterion, the vertex of a triangle to the mid point of the op- posite side is called ‘median’ two triangles in the figure are congruent? Statement (B): The point of concurrence of the altitudes of a triangle is called ‘Ortho center’. (A) Both A and B are true (B) Both A and B are false 61

PRACTICE SHEET - 2 (PS-2) (C) A is false, B is true (D)A is true, B is false II. Short answer questions. 1. In the given figure, AB || CD show that ∅AOB ~ ∅DOC . 2. Apply the appropriate Axiom (Rule) find x and y from figure. 3. Compare and identify ∅PQR, ∅SQR are congruent, it PQ = SR and ∠PQR = ∠QRS III. Long answer questions. 1. In the given figure, DA ⊥ AB, CB ⊥ AB and OM ⊥ AB . If AO = 5.4 cm, OC = 7.2 cm and BO = 6 cm, the find the length of DO. 2. ABCD is parallelogram AC is one of its diagonal. Analyse the figure and explain the relation between ABC and CDA? 62

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. What is the criterion for any 2 plane figures to 5. In two triangles, two angles and a side of the be congruent? (1 Mark) first triangle are equal to the two angles and a side of the second triangle. Will these two triangles always be congruent? (3 Marks) 2. If ∆PEN = ∆CAR , PN = CR then name all the other corresponding parts of ∆PEN and ∆CAR. (2 Marks) 3. If all the sides of a triangle are equal to the sides of another triangle, will both the triangles be congruent to each other? (1 Mark) 4. Observe the given triangles and explain why ∆ABC ≅ ∆FED ? (2 Marks) A FE BC F 63

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 6. Given EB = BD, AE = CB, ∠A = ∠C = 90°. Is 7. ABC is an isosceles triangle with AB = AC . Prove ∆ABE ≅ ∆CDB congruent? Justify. (3 Marks) that i) ∆ADB ≅ ADC  ii) ∠BAD = ∠CAD D iii) BD = CD  (3 Marks) E A A BC B DC 64

8. Comparing Quantities Learning Outcomes At the end of this chapter, students will be able to: • Convert decimals and fractions to percentages • Compare two quantities in terms of fractions, ratios and vice-versa. and percentages. • Interpret percentages. • Determine equivalent ratios. • Calculate profit, loss and simple interest. • Analyse proportions. Concept Map Key Points • Fractional numbers can be converted into percentages and vice-versa. • In our daily life there are many occasions where we compare 2 quantities. To compare 2 quantities, the • Decimals can be converted into percentages and units of both the quantities in comparison must be vice-versa. the same. • Percentages are widely used in our daily life, • To compare 2 ratios, first convert them into like o Exact number can be determined when a certain fractions. If these like fractions are equal, we say percent of the total quantity is given. the given ratios are equivalent. o Quantities given in ratios can also be converter to %. • If 2 ratios are equivalent, then the 4 quantities in o Increase or decrease in quantity can be expressed the ratio are said to be in proportion. in %. o Profit/loss incurred can be expressed in %. • Percentage is another way of comparing quantities. o While computing interest on amount borrowed, • Percentages are numerator of fractions with rate of interest is given in terms of %. denominator 100. It is denoted by ‘%’ and means ‘hundredths’ too. • To calculate percentage of an item when the total number of items do not add up to 100 , then convert the fraction to an equivalent fraction with denominator 100. 65

8. Comparing Quantities Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEETS • Introduction PS – 1 • Equivalent ratios • Proportion PS – 2 PS – 3 Comparing Quantities • Fraction to % and vice versa PS – 4 • Decimal to % and vice versa PS – 5 Self-Evaluation Sheet • Word problems on % • Profit, loss and simple interest Worksheet for “Comparing Quantities” Evaluation with Self Check or Peer ---- Check* 66

PRACTICE SHEET - 1 (PS-1) 1. Find the ratio of: a. Rs. 10 to 10 paisa b. 8 m to 24 cm c. 20 days to 24 hours d. 4 kg to 240 g 2. A group of 36 students went for a picnic. If 6 students can play with 3 balls, how many balls are required for the group of 36 students? 3. One day Rishi travelled 6 km whereas Soham travelled only 5000 m. What will be the ratio of their distances travelled? 4. Are these ratios equivalent? 4 drivers: 1 hour 20 drivers: 5 hours 5. Cost of 3 dozen bananas is Rs. 90. How many bananas can be purchased for Rs. 30? 6. For the given data, in which year did the team perform better? Year Wins Loses 2017 35 10 2018 45 5 7. A bike travels 350 km with 10 liters petrol. How much petrol is needed to travel 7000 km? 8. For the below given data of marks score each year, answer the questions that follow. English Hindi Mathematics Social Science 2017 75 65 85 93 89 2018 78 80 98 92 92 2019 85 82 100 94 88 a. Find the ratio of total marks scored in the year 2017 and 2019. b. What is the ratio of marks score in Mathematics and Science in the year 2018 and 2019? Are they equivalent? c. Find ratio of marks scored in Social and Science in year 2017 and 2018. d. What is the ratio of marks scored in languages in all years? e. Which year was the highest marks scored? 9. Convert the following ratios into fractions and compare a. 10:8 and 5:4 b. 3:4 and 75:125 c. 5:9 and 45:75 67

PRACTICE SHEET - 2 (PS-2) 1. Convert the given fractional numbers to percents. a. 1 6 b. 9 40 c. 10 7 2. Convert the given decimal fractions to percents. a. 0.75 b. 45.78 c. 0.055 3. What percent represents the shaded region in the figure? 4. What percent represents the unshaded region in the figure? 5. Find the value of z if a. 9% of z is 180 b. 15% of z km is 600 km c. 60% of z is 18 minutes d. 12% of z is 36 liters e. 20% of z is 4.5hrs 6. Solve the following a. 18% of 360 b. 20% of an hour c. 60% of 3350 kg d. 28% of 7500 km e. 23% of 2300 7. Convert the given percent to decimal fractions and to fractions in simplest form: a. 35% b. 75% c. 225% d. 5% e. 9% f. 29% 68

PRACTICE SHEET - 3 (PS-3) 1. Out of 700 students, 120 got pencils and 180 got pens. What percentage of students didn’t get anything? 2. In a school, number of students is 2500. Out of which 35% are girls. How many boys are there in school? 3. The school football team played 32 matches out of which 25% of matches were won. How many matches did the team lose? 4. Ram invested money during the period April 2018 to May 2019 at rate of 12% per annum. If interest he earned is Rs. 1620, find the money invested. 5. The population of a village is 10000. Out of these, 80% of them are literate and of these people, 40% are women. Find the ratio of the number of literate women to the total population. 6. 700 kg of cement mixture contains 45% sand, 35% pure cement and rest stone. What is the mass of stone in cement mixture? 7. A traveller travels 70 km by car, 100 km bike and 230 km by train. Find what percent of total journey he travelled by bike, car, and train. 69

PRACTICE SHEET - 4 (PS-4) 1. A man bought a car at Rs. 7,60,000. After using for few years, he sold it for Rs. 3,80,000. Calculate the percentage of loss. 2. A shopkeeper sold a furniture at Rs. 1360 and made a loss of 15%. Find the cost at which he bought it. 3. Calculate the profit or loss in the following transactions. Also find profit or loss percent in each case. a. A bike is bought at Rs. 2,25,000 and sold at Rs. 1,35,000. b. A dress is bought at Rs. 750 and sold at Rs. 850. c. A land rate was Rs. 3375 per square feet in year 2009 and in year 2010 it is Rs. 4500 per square feet. 4. Convert each part of the ratio to percentage: a. 4:5 b. 3:5:7 c. 3:5 d. 4:7:9 5. Maria buys books at Rs. 375 and sells it at a profit of 25%. How much does she sell it for? 6. Find the amount to be paid at the end of 5 years for the following. a. Principal = Rs. 1,50,000 at 12% p.a. b. Principal = Rs. 3,00,000 at 8.5% p.a. 7. Rs. 9000 becomes Rs. 18000 at simple interest in 8 years. Find the rate percent per annum. 8. Find the principal when amount becomes Rs. 30000 in 5 years at 10% rate of interest. 9. In what time frame does principal of Rs. 25000 become an amount of Rs. 50000 with rate of 10% p.a. 10. Calculate the amount for principal Rs. 3350 at the rate of 15% p.a. in 1 1 years. 2 70

PRACTICE SHEET - 5 (PS-5) I. Choose the correct option: II. Short Answer Questions: 1. In a:b, a is called: 1. 800 kg of mortar consists of 55% sand, 33% (A) Antecedent (B) Consequent (C) First term (D) Both A, C cement and rest lime. Show the mass of lime = 2. The method in which we first find the value 96 kg. of one unit and then the value of the required 2. In a science lab, a DNA model was built using number of units is called: scale 2 cm:0.0000001 mm. If the model of the (A) Multiplication method (B) Cross method DNA chain is 17 cm long, what is the length of (C) Unitary method (D) Both A, B actual chain? 3. Equality of ratio is called: 3. Rama purchased 2 pens for 20 ₹ each. She sold (A) Comparison (B) Proportion one pen for 10% profit and another pen for 10% (C) Consequent (D) Can't say loss. Analyse whether she got loss or profit. 4. If =1 =x y% , then the value of 'y' is: III. Long Answer Questions: 20 100 1. ₹ 9,000 becomes ₹ 18,000 at simple interest in 8 (A) 100 (B) 20 years. Find the rate of interest. (C) 10 (D) 15 2. The pieces of a tangram have been rearranged to 5. 225% is equal to: make the given shape. (A) 9:4 (B) 4:9 (C) 3:2 (D) 2:3 6. 1 = ______ % 6 (A) 16 2 % (B) 16 1 % 3 3 (C) 33 1% (D) 33 2 % 7. If 2 1 3 7 1 :18 , 3 :x= then = _____. 22 x R - Red (A) 2 (B) 3 G - Green (C) 4 (D) 6 B - Blue 8. If SP = ₹ 720 and discount = ₹ 72, then M.P = Analyse the given shape, Answer the following ₹____. questions. (A) ₹ 648 (B) ₹ 792 (C) 10% (D) 110% (i) What percent of total has been coloured: a) Red = _______ 9. Statement (A): Two simple ratios are expressed b) Blue = _______ in the form of single ratio is called compound c) Green = _______ ratio. (ii) Will all these colorus make 100%? Statement (B): a : b and c : d are any two ratios, then their compand ratio is ac : bd. (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. Statement (A): SI = PTR 100 Statement (B): CI = P 1+ R n 100  (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 71

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. Find the ratio of 5 km to 500 cm. (2 Marks) 6. A certain sum of money amounts to Rs.1008 in 2 years and to Rs.1164 in 3.5 years. Find the sum and rate of interest. (3 Marks) 2. Find the whole quantity when 18% of it is 180. (1 Marks) 3. Find the rate of interest when principal of Rs. 10000 get an amount of Rs. 35000 in 5 years. (2 Marks) 4. A television was bought at Rs. 1,56,000 and sold 7. In a school of 1000 student, 34% of students like at Rs. 1,00,000. What is the percent of profit/loss cricket, 24% like football and rest like hockey. made. (2 Marks) Find the number of students who like cricket, football and hockey. Also find the ratio of students who like cricket and football. (3 Marks) 5. Solve the following. (3 Marks) a. 25% of 150 b. 12% of W is 240 c. 13% of W is 390 72

9. Rational Numbers Learning Outcome • Represent rational numbers on number line. • Perform algebraic operations on rational By the end of this chapter, a student will be able to: • Define rational numbers. numbers. • Compare rational numbers. Concept Map Key Points • When a rational number has a denominator that is a positive integer and the numerator and • The number expressed in the form p , in which p denominator have no common factor other than q 1, the rational number is said to be in the standard form. and q are integers and q ≠ 0 is known as a rational number. All fractions and integers are rational • While subtracting two rational numbers, the numbers -0.1,2.75,0.001 and 6 are a few examples additive inverse of the rational number to be oraftriaotnioanl naul nmubmebrsearsr.eA: few characteristics of subtracted is added to the other rational number. • Rational numbers are of two types: positive and negative rational numbers. When both the • Two rational numbers with the same denominator numerator and denominator is a positive integer, can be added by adding the numerators keeping then it is said to be a positive rational number. the denominators same. Two rational numbers When either the numerator or the denominator is with different denominators are added by taking a negative integer, then it is said to be a negative the LCM of the two denominators first and rational number. then converting both the rational numbers to • The number 0 is neither a negative nor a positive their equivalent forms keeping the LCM as the rational number. denominator. • If the numerator and denominator of a rational number are divided or multiplied by a non- • While subtracting, the additive inverse of the zero integer, we get a rational number that is rational number to be subtracted is added to the equivalent to the given rational number. other rational number. • Multiplication of two rational numbers is done by multiplying their numerators and denominators 73

9. Rational Numbers separately and the obtained product will be of the form p . q • Division of one rational number by a non-zero rational number is done by multiplying the rational number by the reciprocal of the other. Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Pre-requisite knowledge, • Revision of number system, Definition and PS –1 Numbers, fractions explanation of Rational Numbers, Positive and Negative Rational Numbers, Rational Numbers in standard form Rational numbers • Comparison of rational number, Rational PS –1 number between two given rational numbers Operations on Rational • Addition of Rational Numbers, PS – 2 Numbers Subtraction of Rational numbers Operations on Rational • Multiplication of Rational Numbers, PS – 3 Numbers Division of Rational Numbers PS - 4 Self-Evaluation Sheet Worksheet for “Rational Numbers” Evaluation with Self Check or ---- Peer Check* 74

PRACTICE SHEET - 1 (PS-1) 1. Write three more rational numbers in each of the following: (i) 2 , 4 , 6 ⊃....... (ii) −8 , −24 , −32 … 10 15 10 30 40 5 2. Give four rational numbers equivalent to: (i) 54  (ii) −72 (iii) −3 3. List five rational numbers between: (i) −1 and 2 (ii) −3 and 4 55 (iii) −3 and −4 77 (iv) 45 and 4 3 4. Rewrite the rational numbers in simplest form: (i) −2250  (ii) −2281 (iii) 50 60 5. Compare the rational numbers: (i) −73 and −6 14 (ii) 8 and 10 40 32 (iii) −250� and −1 4 (iv) −7 and 8 45 (v) −3 and −7 73 6. Arrange in ascending order: (i) 3 , 4 , 5  (ii)  −1 , 2 , −2 536 253 75

PRACTICE SHEET - 2 (PS-2) 1. Find the sum of: (i) 12 + 2  (ii) −2 + 4 3 57 (iii) 8 + 2  (iv) −5 + 7 −5 7 8 2. Find product of: (i) 1 × 8  (ii) 2 × −5 33 53 (iii) −5 × 3  (iv) −7 × −3 75 95 (v) 8 × −5 9 3. Divide: (i) 5 ÷ 3 (ii) 6 ÷ 3 65 7 −14 (iii) −7 ÷ 5 (iv) 3 ÷ −5 96 7 4. Find the value of: (i) 12 − 5 (ii) 3 − −5 7 5 4 (iii) −5 − −3 (iv) −2 − 3 75 34 76

PRACTICE SHEET - 3 (PS-3) 1. Represent 3/4 on the number line. 2. Represent 4/5 on the number line. 3. Represent 11/4 on the number line. 4. Represent 7/2 on the number line. 5. Represent -2/3 on the number line. 6. Represent -5/6 on the number line. 7. Represent -9/5 on the number line. 77

PRACTICE SHEET - 4 (PS-4) I. Choose the correct option: 8. The periodicity of 14 in decimal form is: 1. The value of a proper fraction is always: 49 (A) =1 (B) >1 (C) <1 (D)<-1 (A) 285714 (B) 4 (C) 2857 (D) 6 2. The rational number between any two rational 9. Assertion(A): The multiplicative inverse of a numbers ‘a’, ’b’ and its equidistance from a and b is: non-zero rational number a is b (a ≠ 0 & b ≠ 0) (A)  a  +  b  (B)  a  -  b  ba  2   2   2   2  Reason(R):  a  ×  b  =  b  ×  a  =1 1  b   a   a   b  (C)  2  ( a − b) (D)  1  ( a + b)    2  (a ≠ 0 & b ≠ 0) (A) Both A and R are correct and R is correct ex- 3. The product of a non-zero number and its multiplicative inverse is: planation of A (B) Both A and R are correct and R is not correct (A) -1 (B) 0 explanation of A (C) 1 (D) undefined (C) A is correct, R is incorrect (D) A is incorrect, R is correct 4. Standard form of 2.2 is: (A) 22 (B) 11 10. Statement A: The reciprocal of a positive 10 5 rational number is positive Statement B: The reciprocal of a negative (C) 10 (D) 5 rational number is positive 22 11 (A) Both A & B are False (B) A is true, B is false (C) Both A & B are True (D) A is false, B is true 5.  a  +  b  =  c   c  II. Short Answer Questions: 1. Show 0.01010101……. in p/q form. (a +b) (B) (a + b) (A) (c + c) c (C) (a +b) 2. Solve  −7  +  3  −  2 + 1  +  3 + 1  (ac + bc)  6   8   4   2  c2 (D) c 3. Divide the sum of 65 and 3 by their difference. 6. The value of (-2/3) × [(9/8)÷(3/4)] is: 12 8 (A) -1 (B) 1 (C) 2 (D) -2 III. Long Answer Questions.: 1. In a school 5 of students are boys. If there are 7. The value of  −2009  ×  −2010  ×  −2011  8 is:  2010   2011   2012  240 girls, find the number of boys in the school. (A) −2010 −2009 2. Verify the property a × (b+c) = (a × b) + (a × c) 2011 (B) 2010 on rational numbers by using (C) −2010 −2009 a = -1, b = 2 and c = -3 . 2011 2012 5 15 10 (D) 78

Self-Evaluation Sheet Marks: 15 Time: 30 Mins (1 Mark) 6. Add 1 + 5 + 7 . (3 Marks) 1. Simplify to lowest form −20 . 25 365 2. Simplify: −3 × 5 . (1 Mark) 48 3. Write two rational numbers equivalent to −2 . 7. Find the value of −3 + 4 + −2 . (3 Marks) 5 573 (2 Marks) 4. Compare 3 , 5 . (2 Marks) 56 5. Simplify: −3 × 4 ÷ 5 . (3 Marks) 46 79

10. Practical Geometry Learning Outcomes • Construct triangles when the measure of 2 of the angles and the length of side included between At the end of this chapter, students will be able to: them are known. • Construct parallel lines using compass. • Construct triangles when lengths of 3 sides are • Construct right angled triangle when length of one leg and its hypotenuse are given. known. • Construct triangles when lengths of 2 sides and measure of angle between them are known. Concept Map Key Points • Given a line l and a point P not on it, a line parallel to l and through the point P an be drawn using the idea of equal alternate angles. • Construction of triangles are possible under the following cases:- o Given three side lengths of a triangle (SSS) o Given any two side lengths and measure of angle between these sides (SAS) o Given measure of 2 angles and length of side included between them (ASA) o Given length of hypotenuse and length of one of the legs of a right angled triangle (RHS) • The concept of congruence of triangles is used indirectly. 80

10. Practical Geometry Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET • Construction of Parallel Lines PS – 1 • Construction of triangles – SSS criterion PS – 2 Practical Geometry • Construction of triangles – SAS criterion PS – 3 • Construction of triangles – ASA criterion PS – 4 • Construction of triangles – RHS criterion PS – 5 Worksheet for “Practical Geometry” PS – 6 Evaluation with Self Check or ---- Self-Evaluation Sheet Peer Check* 81

PRACTICE SHEET - 1 (PS-1) 1. Construct a line parallel to the one below that passes through point P. P 2. Construct a line parallel to AB through Q and another line parallel to CD also through Q . What is the figure obtained? D Q A B C 3. Draw a line l. Draw a perpendicular to l at any point on l. On this perpendicular choose a point X, 6 cm away from l. Through X, draw a line m parallel to l. PRACTICE SHEET - 2 (PS-2) 1. Construct a triangle DEF in which DE = 3.2 cm, EF = 3 cm and DF = 3.5 cm. 2. Construct a triangle ABC in which BC = 4 cm, AB = 2 cm and AC = 5 cm. Measure angle B. 3. Construct an equilateral triangle of side 4 cm. Measure ∠A. 4. Draw a triangle PQR with PQ = QR = 5 cm and PR = 4 cm. What type of triangle is this? 82

PRACTICE SHEET - 3 (PS-3) 1. Construct a triangle ABC with AB = 7 cm, BC = 5 cm and ∠B = 50°. 2. Construct a triangle ABC with AB = 4.5 cm and AC = 4 cm and ∠A = 60° . 3. Construct an isosceles triangle in which length of each of its equal side is 5 cm and the angle between them is 105ᵒ 4. Construct ∆ABC such that AB = 5 cm, BC = 3 cm and ∠ABC = 90° . PRACTICE SHEET - 4 (PS-4) 1. Construct ∆PQR given ∠P = 60° , ∠Q = 50° and PQ = 6 cm. 2. Construct ∆ABC if AB = 6 cm, ∠CAG = 105° and ∠ACB = 30° . 3. Examine whether you can construct ∆PQR such that PQ = 8 cm, ∠P = 120°, ∠Q = 60°. Justify your answer. 4. Construct ∆MNO, if MN = 3 cm, ∠M = 90° and ∠N = 30°. PRACTICE SHEET - 5 (PS-5) 1. Construct the right angled triangle ABC where ∠B = 90°, BC = 3 cm and AC = 5 cm. 2. Construct a right angled triangle whose hypotenuse is 10 cm and one leg is 8 cm. 83

PRACTICE SHEET - 6 (PS-6) I. Choose the correct option: 1. For constructing an equilateral triangle how 10. Statement (A): The sum of the three sides of a many measurements needed? (A) 4 (B) 3 triangle is less than the sum of its three (C) 2 (D) 1 medians. Statement (B): The difference of two sides of a 2. The semi-circle shaped instrument in geometry box is called: triangle is equal to the third side (A) Protractor (B) Scale (A) Both A and B are false (C) Compass (D) Set square (B) Both A and B are true 3. An amount of rotation is called: (C) A is true, B is false (A) Degree (B) Angle (C) Measurement (D) All the above (D) A is false, B is true 4. A triangle can be constructed by taking two of II. Short Answer Questions: its angles as: 1. Construct a right-angled isosceles triangle with (A) 110º, 40º (B) 70º, 115º (C) 115º, 90º (D) 90º, 90º one side (other then hypotenuse) of length 4.5 cm. 5. In which of the following cases, a unique trian- 2. Side AB = 6cm, ∠A = 60º , ∠ B = 60º , show that gle can be drawn: (A) AB = 4cm, BC = 8cm, CA = 2 cm given measurements forms an equilateral triangle. (B) BC = 5.2 cm, ∠B = 90º , ∠C = 110º 3. Simplify and construct ∠U , if (C) XY = 5 cm, ∠X = 45º , ∠Y = 60º (D) An isosceles triangle with the length of each ∠T = 60º , ∠S = 50º and TU = 7.5 cm. side 7.2 cm III. Long Answer Questions: 1. Which type of triangle can we form, if base BC = 6. In right angled isosceles triangle ABC, ∠B = 90º , then ∠A and ∠C is: 10 cm, base angle = 60º (∠B) and AB + AC = 12 (A) 30º, 60º (B) 60º, 30º (C) 45º, 45º (D) 90º, 0º cm? 7. What are the situations for constructing a 2. Construct ∆PEN in which PE = 5 cm, ∠E = 45º triangle? and EN = 6 cm. Simplify centroid from above (A) Three angles of the triangle construction. (B) Two sides and one angle (C) Two angles and one side 84 (D) Three sides of the triangle 8. The lengths of the sides of some triangle are given in the answers for this question. Which set of lengths does not make a right-angled triangle? (A) 5 cm, 12 cm, 13 cm (B) 5 cm, 8 cm, 10 cm (C) 3 cm, 4 cm, 5 cm (D) 7 cm, 24 cm, 25 cm 9. Statement (A): If we know only its altitude we can construct equilateral triangle. Statement (B): In triangle ABC, altitude BE= alti- tude CF. Then triangle ABC is isosceles triangle. (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

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. With the length of 3 sides given below, can a 4. Construct a ∆ABC where AB = 5 cm, AC = 6 cm triangle be constructed? and ∠CAB = 60°. (3 Marks) 9 m, 5 m, 3 m (2 Marks) 2. With one side of 9 cm length and ∠A = 90° and ∠B = 110° , construct ∆ABC. (1 Mark) 5. Construct a triangle whose side length AB is 5 m, ∠BAC = 45° and ∠CBA = 50° . (3 Marks) 3. Given a line l and a point M on it, draw a perpendicular MP to l where MP = 3.2 cm and a line q parallel to l through P. (3 Marks) 85

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 6. You are given the measurement of 2 sides of the right angled triangle such that these sides enclose the right angle. What else do you need in order to draw a triangle? (1 Mark) 7. If there are 4 line segments of lengths 3 cm, 7 cm, 14 cm and 21 cm, how many triangles can be drawn from these segments? (2 Marks) 86

11. Perimeter and Area Learning Outcomes • Solve application numerical based on perimeter and area of squares, rectangles, parallelograms and circles. At the end of this chapter, students will be able to: • Find the perimeter and area of squares and triangles. • Find the area of triangles and parallelograms. • Find the perimeter and area of circular objects. Concept Map Key Points • Perimeter is the distance around the closed figure. • Area is the part of the plane occupied by a closed figure. • Formulae for area and plane figures Plane figure Area Perimeter Square Side × Side 4 × Side Rectangle Length × 2 × (Length + Breadth Breadth) Circle π × diameter 2πr π × radius2 Triangle Sum of length of 1 × base × height the three sides Parallelogram 2 2 × (Base + Side) Base × Height • Units of area can be converted as 1cm2 = 100 mm2 1m2 = 10000 cm2 1hectare = 10000 m2 • For finding the cost, area or perimeter (as the case) is multiplied by the rate of the work. 87