International Math Olympiad Sample Paper: Grade 8

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International Math Olympiad Sample Paper

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IMO 2019 - GRADE 8 SAMPLE PAPER LOGICAL REASONING 1.

In a certain code, TRIPPLE is written as SQHOOKD. How is DISPOSE written in that code? ESOPSID CHRONRD ESHTPTF DSOESPI

2.

Select a figure from amongst the answer figures which will continue the same series as established by the five-figure.

3.

If X is coded as +, Y is coded as –, Z is coded as × and W is coded as / then evaluate 2X4Y7Z4W2Y2 –20 –12 –10 –32

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4.

Pointing to a photograph, Vipul said, “she is the daughter of my grandfather's only child”. How is Vipul related to the girl in the photograph? Brother Uncle Father Cousin

5.

Which of the following Venn diagram correctly illustrates the relationship among the classes: Carrot, Food, Vegetables

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6.

Select the correct mirror image of the figure.

7.

Select a suitable figure from the four options that would complete the figure matrix.

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8.

Find out the alternative figure which contains figure (i) as its part:

figure (i)

9.

One morning Nimmi and Simmi were talking to each other face to face at a crossing. If Simmi’s shadow was exactly to the left of Nimmi, which direction was Nimmi facing? East West North South

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10. Six friends are sitting in a circle and are facing the center of the circle. Deepak is between Preeti and Priya. Pritam is between Mahima and Lara. Preeti and Mahima are opposite to each other. Preeti is at the right of Deepak. Who is the second left of Preeti? Priya Lara Deepak Mahima 11. If the first half of the alphabet is written in the reverse order, which letter will be exactly midway between the 9th letter from the left and the 10th letter from the right end? N O B A 12. In the following figure, find the total number of squares?

12 13 15 16 13. Answer the question referring to the symbol-letter-number sequence given below T: 1 A R @ 5 6 + Z M 3 C U: N $ B D 1 / 8 L 9 Q 7 V: ⇔ H I 9 W 2 4 ⊗ 6 F K What is the difference between the total numbers and total letters, which are used in the series of T, U, and V? Four Eight Three Six

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14. The characters in the given figures follow a certain rule. Identify the rules and find the missing number.

45 47 36 35 15. When the given figure is folded to form a cube then which face is opposite to the face with 1?

2 6 4 5 16. If (4.1268)2 = 17.03, Then find the value of √170300 412.68 4126.8 41.268 4.1268 17. Reema rolls a dice. She wins if a prime number occurs. What is the probability of Reena’s winning? 1 6 1 2 2 3 1 3 www.cuemath.com

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18. Find the cube root of 7

2051 . 5832

18 17 17 1 18 27 1 17 17 1 28 1

19. The solid figure shown below.

It has ______triangular faces, _________ rectangular faces, ________ vertices. 3,6,2 3,2,6 2,3,6 None of the above.

20. The value of

(528+247)2 −(528−247)2 528×247

4 2 1056 494

21. In the given figure, AO and BO are the bisectors of ∠ DAB and angle ∠CBA respectively. If ∠ DAB= ∠ CBA = 80˚, Find the measure of ∠ AOB and ∠ DCB.

100˚, 90˚ 90˚, 100˚ 120˚, 100˚ 90˚, 120˚

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22. The volume of a cube and cuboid is in the ratio 8 : 9. If the volume of the cuboid is 8 m3 more than that of cube, then (a) What is the volume of the cuboid? (b) What is the length of each side of the cube? (a)

(b)

64 m3 81 m3 64 m3 72 m3

4m 3m 6m 4m

23. What should be added to 4𝑥 4 + 5𝑥 3 − 2𝑥 2 + 5𝑥 to get 5𝑥 4 + 11𝑥 − 8? 𝑥 4 + 5𝑥 3 + 3𝑥 2 − 4𝑥 + 8 𝑥 4 − 5𝑥 3 + 2𝑥 2 + 6𝑥 − 8 𝑥 4 − 2𝑥 2 + 5𝑥 + 8 𝑥 4 − 𝑥 3 + 2𝑥 2 − 6𝑥 + 8 24. Eight less than three times a number is two more than five times the same number. Find the number. 5 2 –5 –2 25. If 32009 - 32005 + 32008 + 32006 = K 32005, then the value of K is __________. 100 110 120 130 26. The pie chart below shows the number of students participated in a singing competition. If there were 80 Grade 8 students, how many students attended the singing competition in all?

500 400 600 300 www.cuemath.com

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27. If one number of a Pythagorean triplet is m2 + 1, then the other two numbers are ___________ m, m2 + 1 2m, m2 - 1 m2, m2 - 1 m2, m + 1 28. The difference between the compound interest (compound annually) and simple interest on a certain sum at 10% per annum for 2 years each is 36.7. Find the sum.

₹ 3470 ₹ 6730 ₹ 3670 ₹ 3560 29. Given below are the steps of construction of a quadrilateral PQRS, where PQ = 4.5 cm, QR = 7cm, ∠P = 105°, ∠Q = 60° and ∠S = 120° which of the following step is incorrect? Step 1: Draw PQ = 4.5 cm Step 2: Draw ∠XPQ = 105° at P and ∠YQP = 60° at Q Step 3: With Q as center and radius QR = 7cm, draw an arc to intersect QY at R. Step 4: At R, draw ∠PSR = 120° such that RZ meets PX at S. Step 1 Only Step 2 Only Step 3 Only Step 4 Only

Direction (30-31): The given double-column graph compares two kitchen appliance stores. Use the graph to answer the questions.

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30. Which appliance from store B, has been sold exactly as half as compare to store A? Toaster Grill Oven Blender 31. What is the total number of appliances sold at both the stores? 210 230 190 240 32. The given table shows the cost price of various items available at a showroom. Item

Cost (in ₹)

2 Jeans

3200

1 Shirt

1500

3 T-shirts

2400

What will be the total final cost of these items if the GST is 18%? ₹ 8738 ₹ 8837 ₹ 8378 ₹ 8387 33. The below figure is the field of a farmer. Find the area of the field.

5640 m2 5460 m2 6400 m2 5600 m2 www.cuemath.com

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34. Divide (2x2 + 7x + 6) (x − 2) by x2 − 4. 2𝑥 − 3 𝑥−3 𝑥 + 3 2𝑥 + 3 35. In the figure, MORE is a rectangle of length 35 cm and breadth 14 cm. Find the area of shaded region if the radius of the circle is 16 cm. (Take π =3.14)

313.84 cm2 331.52 cm2 352.84 cm2 361.57 cm2

EVERYDAY MATHEMATICS 36. A group of students decided to collect as many rupees from each member of the group as is the number of members. If the total collection amounts to ₹ 9801, the number of members in the group is? 101 88 98 99 37. What is the probability of selecting a prime number from 1, 2, 3,.....10? 1 5 1 7 3 5 2 5

38. The population of a town is 196000. It increased by 7% in the first year and decreases by 5% in the second year. What is the population of the town at the end of 2nd years? 201234 199234 189234 200234 www.cuemath.com

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39. If a boy walks at 14 km/hr instead of 10 km/hr., he would have walked 20 km more. What is the actual distance traveled by him? 50 km 80 km 70 km 60 km 40. T is able to do a piece of work in 20 hours and U can do the same work in 15 hours if they can work together for 4 hours, what is the fraction of work left? 11 15 7 15 2 11 8 15

41. A trader gives a 20 % additional discount on the discounted price after giving an initial discount of 10 % on the marked price of an item. The final selling price of the item is ₹ 1440. Find out the marked price. ₹ 1000 ₹ 2000 ₹ 1880 ₹ 1600 42. A rectangular park 60 m long and 40 m wide has two concrete crossroads running in the middle of the park and the rest of the park has been used as a lawn. The area of the lawn is 2109 m 2. What is the width of the road? 5m 3m 4m 2m 43. The cost of type 1 rice is ₹ 15 per kg and type 2 is ₹ 20 per kg. If both type 1 and type 2 of rice mixed in the ratio 2 : 3, then what is the price per kg of the mixed variety of rice? ₹ 19 per kg ₹ 16 per kg ₹ 17 per kg ₹ 18 per kg

47 3 27 kg wheat at the rate ₹ 172 per kg, kg apple at the rate ₹ 80 per kg and kg of rice 2 40 5 at ₹ 90 per kg. If She gives a ₹ 1000 note to the shopkeeper, how much change should she get back?

44. Ravina buys

₹ 191.90 ₹ 170.10 ₹ 164.24 ₹ 184.31 www.cuemath.com

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45. Six years ago, the ratio of the ages of Vrishika and Surbhi was 6:5. Four years hence the ratio of their ages will be 11:10, what is Surbhi’s age at present? 17 15 18 16 46. Three coins are tossed simultaneously. What is the probability that the outcome will be? (i) At least 2 head. (ii) 2 tails. (iii) At least 3 tails. (i)

(ii)

(iii)

3 8 1 2 1 2 5 8

3 8 3 8 5 8 3 8

1 4 1 8 3 8 1 4

47. Identify X, Y, and Z.

X Cylinder Octagonal Prism Spherical Prism Octagonal Prism

Y

Z

Tetrahedron Triangular Prism Triangular Prism Tetrahedron

Triangular Prism Square Pyramid Cone Triangular Prism

48. Arrange the following steps in the correct order in the construction of a parallelogram, one of whose sides is 4.4 cm and whose diagonals are 5.6 cm and 7 cm. Step 1: Join OA and OB.

Step 2: Draw AB = 4.4 cm, with A as the center and radius 2.8 cm, draw an arc. Step 3: Produce OA to C, such that OC = AO. Produce OB to D such that OB = OD, join AD, BC, and CD. Step 4: With B as center and radius 3.5 cm, draw another arc, cutting the previous arc at point O. 1, 2, 3, 4 4, 2, 3, 1 2, 4, 1, 3, 2, 1, 3, 4

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49. A soft drink company prepares drinks of three different flavours -X, Y, and Z. The production of three over a period of six years has been expressed in the bar graph provided below. Production of three different flavours X, Y, and Z by a company over the years (in lakh bottles).

The total production of flavour Z in 1997 and 1998 is what percentage of the total production of flavour X in 1995 and 1996? 102.25% 115.57% 133.33% 96.67% 50. In the Given Figure QPR is a triangle and STUV is a parallelogram. Find (i) x (ii) y (iii) z

(i)

(ii)

75° 35° 60° 45° 35° 145° 75° 110°

(iii) 110° 105° 30° 35°

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ANSWERS 1. (B)

2. (C)

3. (C)

4. (A)

5. (A)

6. (D)

7. (D)

8. (D)

9.(C)

10.(A)

11. (D)

12. (C)

13. (C)

14. (A)

15. (A)

16. (A)

17. (B)

18. (B)

19. (C)

20. (A)

21. (A)

22. (D)

23. (B)

24. (C)

25. (B)

26. (B)

27. (B)

28. (C)

29. (D)

30. (B)

31. (D)

32. (C)

33. (D)

34. (D)

35. (A)

36. (D)

37. (D)

38. (B)

39. (A)

40. (D)

41. (B)

42. (B)

43. (D)

44. (A)

45. (D)

46. (B)

47. (B)

48. (C)

49. (C)

50. (A)

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SOLUTIONS 1.

TRIPPLE

⟶ SQHOOKD

DISPOSE ⟶ CHRONRD 2.

3.

X⟶ + Y⟶ Z⟶ × W⟶ / 2X4Y7Z4W2Y2 2+4–7×4/2-2 2+4-7×2-2 2 + 4 - 14 - 2 6 - 14 - 2 6 - 16 - 10

4.

Grandfather

Father (only son) Vipul

Daughter

(Brother)

(Sister)

5.

6.

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7.

8.

Nimmi

9.

South

East

West (Simmi’s Shadow)

North Simmi 10.

11.

9th M L K J I H G F E D C B A N O P Q R S T U V W X Y Z 10th

12. 13. Total Numbers = 12 Total letters

= 15

Difference

= 15-12 =3 www.cuemath.com

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14.

24 + 32 + 40 + 16 = 112

11 + 22 + 33 + 44 = 110

12 + 34 + 23 + ? =114 69 + ? = 114 - 69 ? = 45 15. The opposite face of 1 is 2 16.

(4.1268)2 = 17.03 (4.1268)2 × 10000 = 17.03 × 10000 (4.1268 × 100)2 = 170300 (412.68)2 = 170300 412.68 = √170300

17. Outcomes of a dice 1, 2, 3, 4, 5, 6 Prime number= 2, 3, 5 p(E) = 3/6 =

3

√7

18.

3

√

=

√

2

2051 5832

42875 5832 5×5×5×7×7×7

3

=

1

2×2×2×3×3×3×3×3×3

= 35/18 17 18

=1

19. It has 2 triangular faces, 3 rectangular faces, 6 vertices. www.cuemath.com

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20.

(528 + 247)2 − (528 − 247)2 528 × 247

⇒ 𝑎2 - 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏) ⇒

⇒

⇒

[528 + 247 + 528 −2 47] [528 + 247 − 528 + 247] 528 × 247 [528 + 528] [247 + 247] 528 × 247 1056 × 494 528 × 247

=4 21.

∠DAB= ∠CBA=80° In AOB

∠AOB + ∠OAB + ∠OBA = 180° ⇒ ∠AOB + 40° + 40° = 180° ⇒ ∠AOB = 180°-80° ⇒ ∠AOB = 100° In quadrilateral ABCD

∠ADC + ∠DCB + ∠CBA + ∠DAB = 360° ⇒ 110° + ∠DCB + 80° + 80° = 360° ⇒ 270° + ∠DCB = 360° ⇒ ∠DCB = 360° - 270° ⇒ ∠DCB = 90°

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22. The volume of the cube: Volume of cuboid = 8 : 9 The volume of cuboid= 8 + volume of the cube

⇒ ⇒ ⇒

9𝑥 = 8 + 8𝑥 9𝑥 − 8𝑥 = 8 𝑥=8

Volume of cuboid = 9𝑥 =9 × 8 =72 m

3

The volume of cube = 8𝑥 = 8×8

⇒ ⇒ ⇒ ⇒

= 64 m3 𝑎3 = 64 𝑎=4m

23.

4𝑥 4 + 5𝑥 3 + 2𝑥 2 + 5𝑥 + 𝐴 = 5𝑥 4 + 11𝑥 − 8 𝐴 = 5𝑥 4 + 11𝑥 − 8 − 4𝑥 4 − 5𝑥 3 + 2𝑥 2 − 5𝑥 = 1𝑥 4 − 5𝑥 3 + 2𝑥 2 + 6𝑥 − 8

24.

3𝑥 − 8 = 2 + 5𝑥 ⇒ 5𝑥 − 3𝑥 = −8 − 2 ⇒ 2𝑥 = −10 𝑥 = −5

25.

32009 - 32005 + 32008 + 32006 = K × 32005

⇒ 32005 [34- 30 + 33 + 31] ⇒ 32005 [81 – 1 + 27 + 3] ⇒ 32005 [110] ⇒K

= K × 32005 = K × 32005 = K × 32005 =110

26. Grade 8 students = 80 Grade 8 = 100 - (40% + 5% + 25% + 10%) = 20% Let total students = 𝑥 20% of 𝑥 is 80

⇒

20𝑥 100

= 80 80 × 100

⇒

𝑥=

⇒

𝑥 = 400

20

27. The Pythagorean triplet is 2m, m2+1, m2-1

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28.

For C.I 𝑅 t ) 100 10 2 + 100)

A = P (1 + = P (1

121𝑃

⇒

A=

⇒

CI = A - P =

⇒

C.I =

=

For S.I

⇒

=

100 121𝑃 100

-P

21𝑃 100 𝑃×𝑟×𝑡 𝑃×10×2 100 2𝑃

=

100

10

C.I - S.I = 36.7

⇒

21𝑃

2𝑃

− 10 = 36.7 100

⇒

21𝑃−20𝑃

⇒

100 1𝑃

⇒

100

P

= 36.7 = 36.7 = 3670

29. PQ = 4.5 cm, QR = 7cm, ∠P=105°, ∠Q= 60°, ∠S= 120° Step 1: Draw PQ = 4.5 cm Step 2: Draw ∠XPQ= 105° at P and ∠YQP= 60° at Q Step 3: With Q as center and radius QR = 7cm, draw an arc to intersect QY at R. Step 4: Find ∠R by angle sum property of a quadrilateral 105° + 60° + ∠R + 120° = 360°

∠R = 75° At R, draw the angle of 75°, which intersect PX at S. Such that ∠S= 120° 30. Grill Store A = 40 Store B = 20 31. Total = 40 + 20 + 35 + 15 + 30 + 30 + 40 + 30 = 240 32. Total CP = 3200 + 1500 + 2400 = 7100 ⇒ GST = 18 % ⇒

SP = 7100 + 18/100 × 7100 = 7100 +1278 = 8378

33. Area of the field = 2 × area of square + 2 × area of trapezium + area of the rectangle = 2 × 40 × 40 + 2 × ½ (40 + 50) × 10 + 30 × 50 = 3200 + 900 + 1500 = 5600 m2 www.cuemath.com

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34.

35.

Length =35cm Breadth = 14cm Radius = 16 cm Area of shaded region = area of circle - area of rectangle = πr2 – (l × b) = 3.14 × 16 × 16 - 35 × 14 = 803.84 - 490 = 313.84 cm2 36. Number of members in the graph = √9801 =√3 × 3 × 3 × 3 × 11 × 11 = 3 × 3 × 11 = 99 37. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Prime number = 2, 3, 5, 7 P(E) =

4 10

2

=5

38. Initial population = 196000 www.cuemath.com

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In first-year population increase by 7% New population = 196000 + 7 ÷ 100 × 196000 = 196000 + 13720 = 209720 Population after first year = 209720 In second year, population decreases by 5 % New population = 209720 –

5 100

× 209720

= 209720 - 10486 = 199234 39. Let the actual distance travelled by him is x km When speed is 10 km/hr Time= 𝑥/10 When speed is 14 km/hr and distance=𝑥+20 𝑥 + 20 Time = 14 𝑥 + 20 𝑥 ⇒ = 10 14

⇒ 14𝑥 = 10𝑥 + 200 ⇒14𝑥 -10𝑥 = 200 ⇒ 4𝑥 = 200 𝑥=

200 4

, 𝑥 = 50 km

40. One day work of T = One day work of U =

1 20 1

15 They work together for 4 hours so the work is done by them 4 4 12 + 16 28 14 7

=

20

+

15

=

=

60

Remaining work = 1-

7 15

=

60 30 15 − 7

=

15

=

=

15 8

15

41. Let the market price = ₹ 𝑥 First discount = 10% Then price = 𝑥

10𝑥

90𝑥

− 100 = 100

Then additional discount = 20 % 90𝑥

New price =

20

90𝑥

− 100 × 100 100 72𝑥

= 100 ⇒ ⇒

SP =

72𝑥 100

72𝑥

= 1440 100 1440 × 100

⇒

𝑥=

⇒

𝑥 = ₹ 2000

72

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42.

Area of lawn = 2019 m2 Area of rectangle = 60 × 40 = 2400 m2 Let width of the road = a meter Area of crossroad = 60 × a + 40 × a - a × a = 100a – a2 Area of lawn = area of rectangle- area of crossroad 2109 = 2400 - (100a – a2)

⇒ 100a – a2 = 2400 - 2109 ⇒ 100a - a2 = 291 ⇒ a2 - 100a + 291 = 0 ⇒ a2 - 97a - 3a + 291 = 0 ⇒ a (a - 97) - 3) a- 97) = 0 ⇒ (a - 3) (a - 97) = 0 a= 3

a = 97

A cannot be 97 m because the total length is only 60 m so the width of the road is 3m. 43. Cost of type 1 rice = ₹ 15 per kg Cost of type 2 rice = ₹ 20 per kg In the mixed variety of rice let the type 1 is 2𝑥 and type 2 is 3𝑥 Then the price of mixed variety is 2𝑥 × 15 + 3𝑥 × 20 = 2𝑥 + 3𝑥 30𝑥 + 60𝑥

= =

5𝑥 90𝑥 5𝑥

= 18 per/kg 44. Cost of wheat = Cost of apple =

47 40 3

× 172 = ₹ 202.10

× 180 = ₹ 120

2 27

× 90 = ₹ 486

Cost of rice

=

Total cost

= ₹ 808.10

5

She should get a chance = 1000 - 808.10 = ₹ 191.90 www.cuemath.com

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45. Let 6 years ago Vrishika’s age = 6𝑥 and Surbhi’s age = 5𝑥 So present age of Varishika’s age = 6𝑥 + 6 And present age of Surbhi = 5𝑥 + 6 After four years, Varishika’s age = 6𝑥 + 6 + 4 = 6𝑥 + 10 year And Surbhi’s age = 5𝑥 + 6 + 4 = 5𝑥 + 10 year 6𝑥 + 10 11

=

⇒

5𝑥 + 10 10 ⇒60𝑥 + 100 = 55𝑥 + 110 ⇒ 60𝑥 − 55𝑥 = 110 − 100 ⇒ 5𝑥 = 10

𝑥=

10 5

𝑥=2 Surbhi’s present age

= 5𝑥 + 6 = 5 (2) + 6 = 10 + 6 = 16 years

46. Coin 1

H

H

H

H

T

T

T

T

Coin 2

H

H

T

T

H

H

T

T

Coin 3

H

T

H

T

H

T

H

T

(i) (ii)

4 8 3

(iii)

=

1 2

8 1 8

47. X - Octagonal Prism Y - Triangular Prism Z - Square Pyramid 48. Step 2: Draw AB=4.4 cm, with A as center and radius 2.8 cm draw an arc. Step 4: With B as center and radius 3.5 cm, draw another arc cutting the previous arc at point O. Step 1: Join OA and OB Step 3: Produce OA to C, Such That OC = AO, Produce OB to D, such That OB= OD joins AD, BC, and CD.

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49. Total production of flavour Z in 1997 and 1998 = 60 + 60 = 120 lakh bottles Total production of flavour X in 1195 and 1996 = 50 + 40 = 90 lakh bottles So the percentage is

120 90

× 100

= 133.33 % 50. In ∆ PQR

∠P + ∠R = 110° [Exterior angle property] ⇒ 35 + ∠R = 110° ⇒ ∠R = 75° 𝑥 =∠R

[Corresponding angles]

(i) 𝑥= 75° (ii) RP || VS

(Given )

∠ RPS = ∠ VST [Corresponding angles] ⟹ 𝑦 = 35 ° In the parallelogram STUV,

⇒ ⇒ ⇒ ⇒

∠VST + ∠SVU = 180° 𝑦 + 𝑦 + 𝑧 = 180° 2𝑦 + 𝑧 = 180° 2(35) + 𝑧 = 180° 70 + 𝑧 = 180°

(Sum of adjacent angles of a parallelogram is 180°)

(iii) 𝑧 = 110°

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