Vladimir Luzgin

Math Lessons for Gifted Students

Level E

(grade 7)

Center Impulse

Week-end and evening classes for gifted students grades 5-9
Canada, ON, L4K 1T7, Vaughan (Toronto),
80 Glen Shields Ave., Unit #10.
Phone (416)826-7270
vluzgin@hotmail.com

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Content

Click on the lesson!

Lesson 01.

Lesson 02.

Lesson 03.

Lesson 04.

Lesson 05.

Lesson 06.

Lesson 07.

Lesson 08.

Lesson 09.

Lesson 10.

Lesson 01

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1. Calculate mentally and record your answer. Hint: use the Distributive Laws a(b + c) = ab + ac, a(b - c) = ab - ac.

2. Joan earns $6.40 per hour. She receives an increase in pay of 5%.

a) What is her raise?

b) What is her new rate of pay?

3. If M is a point ⁴/₇ of the way from A to B on the number line shown, what is the number located at M?

 A aaaaaaaaaaaaa M aaaaaaaaa B

_|______________|__________|__

18 aaaaaaaaaaaaaaaaaaaaaaaaa 95

4. Solve the following problems.

a) Ray's family is on their way to visit his uncle in Canada. The markers on the highway signs give distances in kilometers. Ray knows a mile is about 1.609 kilometers. He sees a sign "Toronto - 145 kilometers". Estimate how many miles to Toronto.

b) John has a comparison table of metric and customary measurements. In the table, he finds a kilometer is 0.621 miles. How many miles are between two towns which are 3.5 cm on a map, if the scale of the map reads 1 : 250 000?

5. Study Appendix 2. Apply congruent triangles theorems to solve the following problems.

a) Given: AC = BD, angle (ACB) = angle (BDA). Prove that AD = BC.

b) Given: AE = BE, CE = DE. Prove that triangles ACD and BDC are congruent.

c) Given: AD = BC, angle (BCD) = angle (ADC). Prove that AE = BE.

6. Solve the following problems.

a) Two-fifths of the employees of Acme Co. earn salaries above $ 60,000 annually. Two-thirds of the remainder earns salaries between $30,000 and $60,000. What part of the staff earns below $30,000?

b) David received his allowance on Sunday. He spends ⁷/₂₄ of his allowance on Monday and ⁶/₁₇ of the remainder on Tuesday. What part of his allowance is left for the rest of the week?

7. Predict the next two numbers in each pattern. Find an algebraic expression that generates each of these sequences of numbers.

1) 6, 8, 16, 18, 26, 28, ____, ____.

2) 6, 8, 16, 18, 36, 38, ____, ____.

3) 2, 6, 12, 20, 30, 42, ____, ____.

8. Make the statements true by inserting grouping symbols (parenthesis) and any of the four operations (+, -, x, :).

1)   5   5   5   5   5   =   1.2

2)   5   5   5   5   5   =   1.1

9. Find the perimeter of the given figure.

10. Find the value of the following (do without a calculator and show all your work).

11. Find the missing digits in the problems below.

12. Find the missing term.

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Answers 1. aaa 1) 30. aaaaaaaaaaa 2) 0.08. aaaaaaaaaaa 3) 52.

2. aa a) $0.32 aaaaaaaaaa 2) $6.72.

3. aa 62

4. aaa a) 90 miles aaa b) 5.4 miles.

5.
a)

a # a	a Statement	a Reason
a 1. aaa	a AC = BD.	a Given
a 2. a	a Angle (ACB) = Angle (BDA).	a Given
a 3. a	a Angle (AEC) = Angle (BED).	a Opposite angles
a 4. a	a The triangles AEC and BED are congruent.	a AAS: statements 1, 2, and 3.
a 5. aaa	a AE = BE, CE = DE.	a Statement 4.
a 6. aaa	a AD = BC.	a Statement 5, AD = AE + DE = BE + CE = BC.

b)

a # a	a Statement	a Reason
a 1. aaa	a AE = BE.	a Given
a 2. a	a CE = DE.	a Given
a 3. a	a Angle (AEC) = Angle (BED).	a Opposite angles
a 4. a	a The triangles AEC and BED are congruent.	a SAS: statements 1, 2, and 3.
a 5. aaa	a AC = BD.	a Statement 4.
a 6. aaa	a AD = BC.	a Statements 1 and 2, AD = AE + DE = BE + CE = BC.
a 7. a	a The triangles ACD and BDC are congruent.	a SSS: statements 5 and 6, CD = DC.

c)

a # a	a Statement	a Reason
a 1. aaa	a AD = BC.	a Given
a 2. a	a Angle (BCD) = Angle (ADC).	a Given
a 3. a	a CE = DE.	a Statement 2, ITT, part 2.
a 4. aaa	a AE = BE.	a Statements 1 and 4, AE = AD - DE = BC - BE = BE.

or

a # a	a Statement	a Reason
a 1. aaa	a AD = BC.	a Given
a 2. a	a Angle (BCD) = Angle (ADC).	a Given
a 3. a	a The triangles ACD and BDC are congruent.	a SAS, Statements 1 and 2, CD = DC.
a 4. aaa	a AC = BD.	a Statement 3.
a 5. a	a Angle (CAD) = Angle (DBC).	a Statement 3.
a 6. a	a The triangles AEC and BED are congruent.	a SAS, Statements 1, 4 and 5.
a 1. aaa	a AE = BE.	a Statements 6.

6. aaa a) ¹/₅. aaaaaa b) ¹¹/₂₄

7. aaa 1) 36, 38; aaaaaaaa 2) 76, 78; aaaaaaa 3) 56, 72.

8.

1) a 5 : 5 + 5 : (5 x 5) a = a 1.2

2) a 5 5 : 5 : (5 + 5) a = a 1.1

9. a 44 cm

10. aa 1) 3 536 aaa 2) ¹/₈

11.

12. a 1) 3 a 2) 5 a 3) 10 a 4) 18 a 5) 80 a 6) 2 a 7) 6 a 8) 9

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Lesson 02

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1. Calculate mentally and record your answer. Hint: use the Distributive Laws a(b + c) = ab + ac, a(b - c) = ab - ac.

2. Solve the following problems.

a) The list price of a car that sold for $21 000 was raised by $750. What was the percentage increase in price?

b) The regular price of a jacket is $129.95. The jacket is put on sale for $100. Express the reduction in price as a percent.

3. Points A, B, C, and D lie on a straight line, and B is between A and C, C is between B and D.

a) Find BC if AD = 37 cm, AC = 17 cm, BD = 24 cm.

b) Find AD if BC = 6 cm and AB : BC : CD = 7 : 3 : 5.

4. Solve the following problems.

a) Ray knows the meter corresponds to inches as a unit of length. A meter is 39.37 inches. If Ray wants a rope 100 inches long, how many meters is that, rounded to the nearest hundredth?

b) Ray checks his measurement table and learns a foot is 3.048 decimeters. He knows his sister is 1.56 meters tall. Ray converts that to feet and inches. What is his sister's height, rounded to the nearest inch? (1 foot = 12 inches)

5. Solve the following problems.

a) Given: angle (CAD) = angle (DBC), angle (ABC) = angle (BAD). Prove that CE = DE.

b) Given: AD = AE, AB = AC. Prove that BF = CF.

c) In a triangle ABC, AB = CB, Angle (BCE) = Angle (BAD). Prove that FD = FE.

6. Solve the following problems.

a) The paper reports the price of regular gasoline has increased ⁷/₂₄ in the last year. If the price was $1.20 per gallon last year, what is the price now?

b) Eric reads that fees for large-building construction permits will increase 3 ⁵/₁₂ times. Currently a permit costs $108. What will the new fees be?

7. Predict the next two numbers in each pattern. Find an algebraic expression that generates each of these sequences of numbers.

1) 8, 16, 32, 64, 128, 256, ____, ____.

2) 3, 8, 15, 24, 35, 48, ____, ____.

3) 2, 3, 5, 8, 12, 17, ____, ____.

8. Make the statements true by inserting grouping symbols (parenthesis) and any of the four operations (+, -, x, :).

1)   5   5   5   5   5   =   2.4

2)   5   5   5   5   5   =   0.24

9. Determine the distance around the outside of the given figure.

10. Find the value of the following (do without a calculator and show all your work).

11. Find the missing digits in the problems below.

12. Find the missing term.

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Answers 1. aaa 1) 58. aaaaaaaaaaa 2) 87.

2. aa a) 3.57%. aaaaaaaaaa 2) 23.05%.

3. aa a) 4 cm. aa b) 30 cm.

4. aaa a) 2.54 m aaa b) 5 feet 1 inch.

5.
a)

a # a	a Statement	a Reason
a 1. aaa	a Angle (CAD) = Angle (DBC).	a Given
a 2. a	a Angle (ABC) = Angle (BAD).	a Given
a 3. a	a Angle (ABD) = Angle (BAC).	a Statements 1 and 2, Angle (ABD) = Angle (ABC) + Angle (DBC) a = Angle (BAD) + Angle (CAD) = Angle (BAC).
a 4. a	a The triangles BAC and ABD are congruent.	a ASA: statements 2 and 3, AB = BA.
a 5. aaa	a AC = BD, Angle (ACB) = Angle (BDA).	a Statement 4.
a 6. aaa	a The triangles BAC and ABD are congruent.	a SAS: statements 1 and 5.
a 7. aaa	a CE = DE.	a Statement 6.

b)

a # a	a Statement	a Reason
a 1. aaa	a AD = AE.	a Given
a 2. a	a AB = AC.	a Given
a 3. a	a The triangles ABD and ACE are congruent.	a SAS: statements 1 and 2, Angle (BAD) = Angle (CAE).
a 4. aaa	a Angle (B) = Angle (C).	a Statement 3.
a 5. aaa	a CD = BE.	a Statements 1 and 2, CD = AC - AD = a AB - AE = BE.
a 6. a	a Angle (BFE) = Angle (CFD).	a Opposite Angles.
a 7. a	a The triangles BFE and CFD are congruent.	a AAS: statements 4, 5 and 6.
a 8. a	a FD = FE.	a Statement 7.

c)

a # a	a Statement	a Reason
a 1. aaa	a AB = CB.	a Given
a 2. a	a Angle (BCE) = Angle (BAD).	a Given
a 3. a	a Angle (BAC) = Angle (BCA).	a Statement 1, ITT, part 1.
a 4. a	a Angle (CAF) = Angle (ACF).	a Statements 2 and 3, Angle (CAF) = Angle (BAC) - Angle (BAD) a = Angle (BCA) - Angle (BCE) = Angle (ACF).
a 5. a	a AF = CF.	a Statement 4, ITT, part 2.
a 6. a	a Angle (AFE) = Angle (CFD).	a Opposite Angles.
a 7. a	a The triangles AFE and CFD are congruent.	a ASA: statements 2, 5 and 6.
a 8. a	a FE = FD.	a Statement 7.

6. aaa a) $1.55 per gallon. aaaaaa b) $369.

7. aaa 1) 256 x 2 = 512, 512 x 2 = 1024; aaaaaaaa 2) 8² - 1 = 63, 9² - 1 = 80; aaaaaaa 3) 17 + 6 = 23, 23 + 7 = 30.

8.

1) a (5 5 + 5) : (5 x 5) a = a 2.4

2) a (5 + 5 : 5) : (5 x 5) a = a 0.24

9. a 58 cm

10. aa 1) 0.64 aaaaaaaaaaaaaaa 2) 47.9

11.

12. a 1) 0.9 a 2) 0.5625 a 3) 3 a 4) 18 a 5) 0.5 a 6) 2 ¹/₃ a 7) 5 a 8) 3.28125

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Lesson 03

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1. Calculate mentally and record your answer. Hint: use the Distributive Laws a(b + c) = ab + ac, a(b - c) = ab - ac.

2. Solve the following problems.

a) A car that sold for $18 500, now sells for $25 000. Express the increase in price as a percent.

b) Last year, the Micro Electronics Company had a 2.7 million dollar profit. What is this year's profit if there is an increase of 125% over last year?

3. A, B, C, and D are four points on a line such that AB : AC = 3 : 5 and BC : CD = 7 : 12. If CD is 20 cm long, find the length of AB.

4. Solve the following problems.

a) Ray tells his father, "I'm so thirsty I could drink more than a camel drinks." Ray's measurement table shows a liquid quart to equal 0.946 liter. If a camel can drink 40 liters of water, how many liquid quarts does the camel drink?

b) Ray's uncle owns a farm of 80 hectares. How many acres does Ray's uncle own? An acre equals 0.405 hectares.

5. Solve the following problems.

a) Given: AB and CD bisect each other at E, FG passes through E. Prove that EF = EG.

b) Given: Angle (BAC) = Angle (LKM), AD bisects angle BAC, KN bisects angle LKM, AC = KM, AD = KN. Prove that BD = LN.

c) Given: points A, B, C, and D are collinear and triangles ABE and ABF are congruent. Prove that triangles ECD and FCD are also congruent.

6. Solve the following problems.

a) Eric likes outdoor winter sports. This year's deep snows were delightful for him. A local statistical reports shows the snowfall was 1 ³/₄ times greater than the average of 32.48 inches. How many inches of snowfall were there this year?

b) "I'll pay you $8.25 an hour if you will mow my yard", Joel's neighbor said. "It should take you about 3 hours and 20 minutes." Joel multiplied the numbers and agreed to do it when he found out how much he would make. How much pay would he receive?

7. Predict the next two numbers in each pattern. Find an algebraic expression that generates each of these sequences of numbers.

1) 1, 3, 7, 15, 31, 63, ____, ____.

2) 1, 4, 9, 16, 25, 36, ____, ____.

3) 1, 3, 9, 27, 81, 243, ____, ____.

8. Make the statements true by inserting grouping symbols (parenthesis) and any of the four operations (+, -, x, :).

1)   1   2   3   4   5   =   2.9

2)   1   2   3   4   5   =   0.01

9. Write expressions for the perimeter and the area of the figure below.

10. Find the value of the following (do without a calculator and show all your work).

11. Find the missing digits in the problem below.

12. Find the unknown value in each of the following proportions.

1) x : 51.6 = 11.2 : 34.4 aaaa 2) ⁴/₇ : ²⁰/₂₁ = x : 4 ¹/₆

3) 3 ¹/₅ : x = 2 ¹/₄ : 4 ¹/₂ aaa 4) 2 ⁸/₁₅ : 3 ⁴/₅ = 1.5 : x

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Answers 1. aaa 1) 16.5; aaaaaaaaaaaa 2) 2.4.

2. aa a) 35.14%; aaaaaaaaaa 2) 6.075 million.

3. aa 17.5 cm.

4. aa a) 42.283 quart aaaaaa b) 197.53 acre.

5.
a)

a # a	a Statement	a Reason
a 1. aaa	a AE = BE.	a Given
a 2. aaa	a DE = CE.	a Given
a 3. a	a Angle (AED) = Angle (BEC).	a Opposite angles
a 4. a	a The triangles AED and BEC are congruent.	a SAS: statements 1, 2 and 3.
a 5. aaa	a Angle (EAF) = Angle (EBG).	a Statement 4.
a 6. a	a Angle (AEF) = Angle (BEG).	a Opposite angles
a 7. aaa	a The triangles AFE and BGE are congruent.	a ASA: statements 1, 5 and 6.
a 8. aaa	a EF = EG.	a Statement 7.

b)

a # a	a Statement	a Reason
a 1. aaa	a Angle (BAC) = Angle (LKM).	a Given
a 2. aaa	a AD bisects the angle (BAC).	a Given
a 3. aaa	a KN bisects the angle (LKM).	a Given
a 4. a	a AC = KM.	a Given
a 5. a	a AD = KN.	a Given
a 6. aaa	a Angle (DAC) = Angle (NKM).	a Statements 1, 2 and 3.
a 7. a	a The triangles CAD and MKN are congruent.	a ASA: statements 4, 5 and 6.
a 8. aaa	a Angle (ADC) = Angle (KNM).	a Statement 7.
a 9. aaa	a Angle (ADB) = Angle (KNL).	a Statement 8, Supplementary Angle Theorem.
a 10. aaa	a Angle (BAD) = Angle (LKN).	a Statements 1, 2 and 3.
a 11. a	a The triangles ABD and KLN are congruent.	a ASA: statements 5, 9 and 10.
a 12. aaa	a BD = LN.	a Statement 11.

c)

a # a	a Statement	a Reason
a 1. a	a Points A, B, C, and D are collinear.	a Given.
a 2. a	a The triangles ABE and ABF are congruent.	a Given.
a 3. aaa	a BE = BF.	a Statement 2.
a 4. a	a Angle (ABE) = Angle (ABF).	a Statement 2.
a 5. a	a Angle (EBC) = Angle (FBC).	a Statements 1 and 4, Supplementary Angle Theorem.
a 6. a	a The triangles EBC and FBC are congruent.	a SAS: Statements 3 and 5, BC = BC .
a 7. a	a EC = FC.	a Statement 6.
a 8. a	a Angle (BCE) = Angle (BCF).	a Statement 6.
a 9. a	a Angle (ECD) = Angle (FCD).	a Statements 1 and 8, Supplementary Angle Theorem.
a 10. a	a The triangles ECD and FCD are congruent.	a SAS: statements 7 and 9, CD = CD.

6. aaa a) 56.84 inches. aaaaaa b) $27.5.

7. aaa 1) 63 x 2 + 1 = 127, 127 x 2 + 1 = 255; aaaaaaaa 2) 7² = 49, 8² = 64; aaaaaaa 3) 243 x 3 = 729, 729 x 3 = 2 187.

8.

1) a 1 : 2 + (3 x 4) : 5 a = a 2.9

2) a 1 : [(2 + 3) x 4 x 5] a = a 0.01

9. aa 1) P = 2x + 2y; aaaaaaaaaaaaaaa 2) A = xy - (x - z)(y - u) = zu - xu - yz.

10. aa 1) 2 ⁷/₉; aaa 2) ⁴/_7; aaa 3) 0.5 aaa 4) 0.5.

11.

12. a 1) 16.8; aaa 2) 2.5 aaa 3) 6.4 aaa 4) 2.25

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Lesson 04

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1. Calculate mentally and record your answer.

2. Greta earns $16.50/h and receives a raise of 4.2%.

a) What is the hourly raise in pay?

b) What is the new hourly rate of pay?

3. Solve the following problems.

a) Two numbers are in the ratio 4 to 7. Their sum is 132. Find the numbers.

b) Two numbers are in the ratio of 7 : 3. The difference of the two numbers is 24. Find the numbers.

4. Ray tells his sister, "The metric system is easy because it is based on decimals." If one customary gallon is 3.7852 liters, how many gallons are 10 liters?

5. Prove the following theorems that provide sufficient condicians to have congruent right-angled triangles.

(LL) Leg - Leg Theorem. If two legs of one right-angled triangle are respectively equal to two legs of another right-angled triangle, then the triangles are congruent.

(HA) Hypotenuse - Acute Angle Theorem. If the hypotenuse and an acute angle of one right-angled triangle are respectively equal to the hypotenuse and an acute angle of another right-angled triangle, then the triangles are congruent.

(LA) Leg - Acute Angle Theorem. If a leg and an acute angle of one right-angled triangle are respectively equal to a leg and an acute angle of another right-angled triangle, then the triangles are congruent.

6. Solve the following problems.

a) The group decided to buy two bottles of drinks for $1.08 each. Andrew agreed to pay ²/₉ and his three friends would equally pay for the rest. How much did each of the friends pay?

b) Seth bought 19 ⁵/₆ pounds of peanuts. He gave ³/₇ of his purchase to his sister. How many pounds of peanuts did Seth keep for himself?

7. Predict the next two numbers in each pattern. Find an algebraic expression that generates each of these sequences of numbers.

1) 1, 1, 2, 3, 5, 8, ____, ____.

2) 1, 2, 6, 24, 120, 720, ____, ____.

3) 1, 8, 9, 27, 64, 125, ____, ____.

8. Make the statements true by inserting grouping symbols (parenthesis) and any of the four operations (+, -, x, :).

1)   1   2   3   4   5   =   0.2

2)   1   2   3   4   5   =   0.16

9. What is the perimeter of the figure ABCDEFGHIJ, if AB = 14 cm, AJ = 6 cm, GF = 3 cm, and ED = 4 cm?

10. Find the value of the following (do without a calculator and show all your work).

11. Find the missing digits in the problem below.

12. Find the unknown value in each of the following proportions.

1) 3 ⁹/₁₄ : 2 ¹/₇ = x : 1.5 aaa 2) 4 ²/₅ : x = 8 ⁴/₅ : 2 ¹/₂

3) (x - 3) : 6 = 7 : 3 aaaaaaa 4) 3 : 7 = (2 - x) : 5

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Answers 1. aaa 1) ⁴/₃₅; aaaaaaaaaaaa 2) 0.8.

2. aa a) $0.69; aaaaaaaaaa 2) $17.19.

3. aaa 1) 48, 84; aaa 2) 42, 18.

4. aa 2.64 gallons

5.
(LL)

a # a	a Statement	a Reason
a 1. aaa	a AC = KM.	a Given
a 2. aaa	a BC = LM.	a Given
a 3. a	a Angle (ACB) = Angle (KML) = 90o.	a Given
a 4. a	a The triangles ACB and KLM are congruent.	a SAS: statements 1, 2 and 3.

(HA)

a # a	a Statement	a Reason
a 1. aaa	a Angle (BAC) = Angle (LKM).	a Given
a 2. aaa	a Angle (ACB) = Angle (KML) = 90o.	a Given
a 3. a	a AB = KL.	a Given
a 4. aaa	a Angle (ABC) = Angle (KLM).	a Statements 1, 2, Complementary Angles Theorem.
a 5. a	a The triangles ABC and KLM are congruent.	a ASA: statements 1, 3 and 4.

(LA)

a # a	a Statement	a Reason
a 1. aaa	a Angle (BAC) = Angle (LKM).	a Given
a 2. aaa	a Angle (ACB) = Angle (KML) = 90o.	a Given
a 3. a	a AC = KM.	a Given
a 4. a	a The triangles ABC and KLM are congruent.	a ASA: statements 1, 2 and 3.

6. aaa a) $0.56; aaaaaa b) 11¹/₃ pounds.

7. aaa 1) 13, 21; aaaaaaaa 2) 5040, 40320; aaaaaaa 3) 216, 343.

8.

1) a 1 x (2 + 3 - 4) : 5 a = a 0.2

2) a 1 : (2 + 3) x 4 : 5 a = a 0.16

9. aa 54 cm.

10. aa 1) 3; aaa 2) 3; aaa 3) ⁴⁹/₂₁₆ aaa 4) ³/₈.

11.

12. a 1) 2.55; aaa 2) 1.25 aaa 3) 17 aaa 4) -¹/₇.

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Lesson 05

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1. Calculate mentally and record your answer.

2. A property was purchased for $80 000 and later sold for $100 000.

(a) Express the increase in value as a percentage of the purchase price.

(b) Express the selling price as a percentage of the purchase price.

3. Solve the following problems.

a) The ratio of boys to girls in a school is 8 : 7. There are 392 boys in the school. How many girls are there?

b) Alex and Fred share a $1 680 prize in the ratio 11 : 13. How much does each receive?

4. Solve the following problems.

a) Ray asks his sister, "If 1 mile is 1.609 kilometers, how many square miles in one square kilometer?" What is the answer?

b) Emily was considering growing tomatoes the next year. She was told that a hectare of tomatoes produces 500 bushels of tomatoes. If she were to plant 2 800 acres of tomatoes, how many bushels would she produce? An acre equals 0.405 hectares.

5. Solve the following problems.

1) ABC is an isosceles triangle with AB = AC. BC is extended to D, and CB is extended to E, BE = CD. Prove that AE = AD.

2) ABC is an isosceles triangle with AB = AC. D and E are points on BC such that BD = CE. Prove that ADE is an issosceles triangle.

6. Solve the following problems.

a) A ball is dropped from 90 cm and always bounces ²/₃ of the height from which it falls. How many centimeters will the ball rise after the third bounce?

b) Anne bought a soccer ball for $24.00. She sold it to Bill for ¹/₆ less than she paid for it. Bill sold it to Cathy for ¹/₅ less than he paid for it. Cathy sold it to Dave for ¹/₄ less than she paid for it. Find the price that Dave paid for the ball.

7. Predict the next two numbers in each pattern. Find an algebraic expression that generates each of these sequences of numbers.

1) 1, 3, 5, 7, 9, 11, ____, ____.

2) 1, 3, 6, 10, 15, 21, ____, ____.

3) 1, 2, 5, 10, 17, 26, ____, ____.

8. Make the statements true by inserting grouping symbols (parenthesis) and any of the four operations (+, -, x, :).

1)   5   4   3   2   =   0.625

2)   5   4   3   2   =   0.2

9. Find expressions for the perimiter and the area of the figure below.

10. Find the value of the following (do without a calculator and show all your work). Round your answer to the nearest hundredth if necessary.

11. Find the missing digits in the problem below. Find two possible solutions.

12. Find the unknown value in each of the following proportions.

1) 0.2 : (x - 2) = 0.5 : 2.5 aaa 2) 4 ¹/₃ : (2x) = 1.3 : 3

3) 4 : 28 = 4.5 : (3x) aaaaaaa 4) (6 - x) : 3 = x : 6

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Answers 1. aaa 1) 0.25; aaaaaaaaaaaa 2) ²/₃.

2. aa a) 25%; aaaaaaaaaa 2) 125%.

3. aaa a) 343; aaa b) Alex: $770, Fred: $910.

4. aa a) 0.386 sq. miles. aa b) 567 000.

5.

1)

a # a	a Statement	a Reason
a 1. aaa	a AB = AC.	a Given
a 2. aaa	a BE = CD.	a Given
a 3. a	a Angle (ABC) = Angle (ACB).	a Statement 1, ITT, part 1.
a 4. a	a Angle (ABE) = Angle (ACD).	a Statement 3, supplementary angles.
a 5. a	a The triangles ABE and ACD are congruent.	a SAS: statements 1, 2 and 4.
a 6. aaa	a AE = AD.	a Statement 5.

2)

a # a	a Statement	a Reason
a 1. aaa	a AB = AC.	a Given
a 2. aaa	a BD = CE.	a Given
a 3. a	a Angle (ABD) = Angle (ACE).	a Statement 1, ITT, part 1.
a 4. a	a The triangles ABD and ACE are congruent.	a SAS: statements 1, 2 and 3.
a 5. a	a AD = AE.	a Statement 4.
a 6. aaa	a ADE is an isosceles triangle.	a Statement 5.

6. aaa a) 26²/₃ cm; aaaaaa b) $12.

7. aaa 1) 13, 15; aaaaaaaa 2) 28, 36; aaaaaaa 3) 37, 50.

8.

1) a 5 x 4 : 32 a = a 0.625

2) a (5 - 4) : (3 + 2) a = a 0.2

9. aa P = 2(x + y + z) cm, A = xy - zu cm².

10. aa 1) 3.07; aaa 2) 68.8.

11.

12. a 1) 3; aaa 2) 5; aaa 3) 10.5; aaa 4) 4.

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Lesson 06

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1. Calculate mentally and record your answer.

2. Solve the following problems.

a) A vehicle can travel 120 km on 12 L of fuel. If the efficiency of the vehicle is increased by 20%, what distance will it now travel on 63.5 L of fuel?

b) The Big Sound Audio Store had sales of $2 525 000 last year. This year, the sales are expected to increase by 14.7%. What is the expected sales figure for this year?

3. Solve the following problems.

a) A piece of rope 2.4 m long is to be cut into 2 pieces in the ratio 3 : 5. How long is the shorter piece?

b) One angle of a triangle is 82^o. The other two angles are in the ratio 2 : 5. Find the number of degrees in the smallest angle of the triangle.

4. Solve the following problems.

a) Under certain conditions sound travels at about 1 100 feet per second. What is the speed of sound in miles per hour? 1 mile = 1760 yards, 1 yard = 3 feet.

b) An airplane flies 550 yards in 3 seconds. What is the speed of the airplane, expressed in miles per hour?

5. ABC is an isosceles triangle with AB = AC.

1) BM and CN are the medians drawn to AC and AB respectively. Prove that BM = CN.

2) Prove that the altitudes BH and CG are equal.

3) Prove that the angular bisectors BD and CE are equal.

6. Solve the following problems.

a) The art crew needed to paint a mural that covered ³/₄ of the length of the back wall and ⁸/₁₅ of the height of the back wall. What part of the whole wall was covered by the mural?

b) Four students share $936. Al and Bob receive ⁷/₂₄ and ⁵/₁₈ of the total, respectively. Carl receives the average of what Al and Bob got. If Dave collects the remaining money, determine how much his share is.

7. Predict the next two numbers in each pattern. What expression creates the number pattern?

1) 1, 4, 7, 10, 13, 16, ____, ____.

2) 3, 8, 13, 18, 23, 28, ____, ____.

3) 11, 19, 27, 35, 43, 51, ____, ____.

8. Make the statements true by inserting grouping symbols (parenthesis) and any of the four operations (+, -, x, :).

1)   5   4   3   2   =   1.75

2)   5   4   3   2   =   0.5

9. A rectangular flower bed, dimensions 16 yards by 12 yards, is surrounded by a walk 3 yards wide. What is the area of the walk in square yards?

10. Find the value of the following (do without a calculator and show all your work).

11. Find the missing digits in the problem below.

12. Calculate the values of x and y in each of the following proportions.

1) x : 1 : 2 = 10 : y : 4 aaa 2) 2 : 3 : 4 = 8 : x : y aaa 3) 2 : x : 3 = 6 : 12 : y

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Answers 1. aaa 1) -22; aaaaaaaaaaaa 2) ¹/₁₆.

2. aa a) 762 km; aaaaaaaaaa 2) $ 2 896 175.

3. aaa a) 0.9 m; aaa b) 28^o.

4. aa a) 750 miles per hour; aa b) 375 miles per hour.

5.

1)

a # a	a Statement	a Reason
a 1. aaa	a AB = AC.	a Given
a 2. aaa	a AN = BN.	a Given (CN is a median).
a 3. aaa	a AM = CM.	a Given (BM is a median).
a 4. aaa	a AM = AN.	a Statements 1, 2, and 3.
a 5. a	a The triangles AMB and ANC are congruent.	a SAS: statements 1 and 4, Angle(BAM) = Angle(CAN).
a 6. aaa	a BM = CN.	a Statement 5.

2)

a # a	a Statement	a Reason
a 1. aaa	a AB = AC.	a Given
a 2. a	a Angle (BGC) = 90o.	a Given (CG is an altitude).
a 3. a	a Angle (CHB) = 90o.	a Given (BH is an altitude).
a 4. a	a The triangles ABH and ACG are congruent.	a HA: statements 1, 2 and 3, Angle(BAH) = Angle(CAG).
a 5. aaa	a BH = CG.	a Statement 4.

3)

a # a	a Statement	a Reason
a 1. aaa	a AB = AC.	a Given
a 2. aaa	a Angle (CBD) = Angle (ABD).	a Given (AD is an angular bisector).
a 3. aaa	a Angle (BCE) = Angle (ACE).	a Given (CE is an angular bisector).
a 4. a	a Angle (ABC) = Angle (ACB).	a Statement 1, ITT, part 1.
a 5. aaa	a Angle (ABD) = Angle (ACE).	a Statements 2, 3, and 4.
a 6. a	a The triangles ABD and ACE are congruent.	a ASA: statements 1 and 5, Angle (BAD) = Angle (CAE).
a 7. aaa	a BD = CE.	a Statement 6.

6. aaa a) ²/₅; aaaaaa b) $136.5.

7. aaa 1) 19, 22, 3n - 2; aaaaaaaa 2) 33, 38, 5n - 2; aaaaaaa 3) 59, 67, 8n + 3.

8.

1) a 5 : 4 x 3 - 2 a = a 1.75

2) a 5 : (4 x 3 - 2) a = a 0.5

9. aa 204 sq. yards.

10. aa 1) 78.7; aaa 2) 1.

11.

12. a 1) x = 5, y = 2; aaa 2) x = 12, y = 16; aaa 3) x = 4, y = 9.

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Lesson 07

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1. Calculate mentally and record your answer.

2. Solve the following problems.

a) The sides of a square are 40 cm. The length of each side is reduced by 25%. What is the percentage change in the area of the square?

b) Harry earns a salary of $360 per week for 44-hour week. His weekly salary is increased by 10% and his hours are reduced by 10%. Calculate his new hourly salary.

3. Solve the following problems.

a) A brass alloy contains cooper and zinc in the ratio 20 : 7. How much cooper is there in 9 kg of this alloy?

b) Propane is a combination of carbon and hydrogen in the ratio of 18 : 4 by mass. What mass of hydrogen is in 165 g of propane?

4. Solve the following problems.

a) Mrs. Graham comes into the carpet store to look for carpet for her foyer. The dimensions of the foyer are 7.2 feet wide and 8.7 feet long. Cheryl shows her various samples of carpet. Mrs. Graham likes the dark blue carpet at $54.99 per square yard. What is the cost of the blue carpet for this area? (1 yard = 3 feet)

b) If it costs $1.30 a square foot to lay linoleum, what will be the cost of laying 20 square yards of linoleum?

5. Prove that if ABC is an isosceles triangle,

1) the median AM is perpendicular to BC;

2) the hight AH bisects the angle BAC;

3) the bisector AD bisects the side BC.

6. Solve the following problems.

a) A team played 30 games of which it won 24. What part of the games played did it lose?

b) In a class of 36 students the number of boys is 6 less than the number of girls. What fraction of the class is female?

7. Predict the next two numbers in each pattern. What expression creates the number pattern?

1) 5, 12, 19, 26, 33, 40, ____, ____.

2) 3, 6, 11, 18, 27, 38, ____, ____.

8. Make the statements true by inserting grouping symbols (parenthesis) and any of the four operations (+, -, x, :).

1)   3   3   3   3   =   1¹/₃

2)   5   5   5   5   =   ²/₁₁

9. In a triangle ABC, AD _|_ BC, BE _|_ AC, AD = 4 cm, BC = 6 cm, AC = 5 cm. Find the length of BE.

10. Find the value of the following (do without a calculator and show all your work).

11. Find the missing digits in the problem below.

12. Calculate the values of x and y in each of the following proportions.

1) 3 : 10 : x = 1 : y : 3 aaa 2) 1 : 3 : 2 = x : y : 8 aaa 3) 7 : 3 : 1 = x : 6 : y

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Answers 1. aaa 1) ¹/₈; aaaaaaaaaaaa 2) 4.

2. aa a) reduced by 43.75%; aaaaaaaaaa 2) $ 10/h.

3. aaa a) 6.667 kg; aaa b) 30 g.

4. aa a) $ 382.73 aa b) $ 234.

5.

1)

a # a	a Statement	a Reason
a 1. aaa	a AB = AC.	a Given
a 2. aaa	a BM = CM.	a Given
a 3. a	a The triangles ABM and ACM are congruent.	a SSS: statements 1 and 2, AM is common.
a 4. aaa	a Angle (AMB) = Angle (AMC) = 90o.	a Statement 3, supplementary angles.
a 5. aaa	a AM _|_ BC	a Statement 4.

2)

a # a	a Statement	a Reason
a 1. aaa	a AB = AC.	a Given
a 2. a	a AH _|_ BC.	a Given
a 3. a	a Angle (AHB) = Angle (AHC) = 90o.	a Given
a 4. a	a The triangles AHB and AHC are congruent.	a HS: statements 1 and 3, BH is common.
a 5. aaa	a Angle (BAH) = Angle (CAH).	a Statement 4.
a 6. aaa	a AH bisects Angle (BAC).	a Statement 5.

3)

a # a	a Statement	a Reason
a 1. aaa	a AB = AC.	a Given
a 2. aaa	a Angle (BAD) = Angle (CAD).	a Given
a 3. a	a The triangles BAD and CAD are congruent.	a ASA: statements 1 and 2, BD is common.
a 4. aaa	a BD = CD.	a Statement 3.

6. aaa a) ¹/₅; aaaaaa b) ⁷/₁₂;

7. aaa 1) 47, 54, 7n - 2; aaaaaaaa 2) 51, 66, n² + 2.

8.

1) a (3 + 3 : 3) : 3 a = a 1¹/₃;

2) a (5 + 5) : 55 a = a ²/₁₁;

9. aa 4.8 cm.

10. aa 1) 6; aaa 2) 45.9.

11.

12. a 1) x = 9, y = 3¹/₃; aaa 2) x = 4, y = 12; aaa 3) x = 14, y = 2.

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Lesson 08

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1. Calculate mentally and record your answer.

2. Solve the following problems.

a) Eric bought a new suit for $98.00. What is the total price he paid if the GST is 7% and PST is 8%?

b) A shirt costs $39.95. Sales tax is charged at a rate of 7%. What is the cost of the shirt, including tax?

3. Solve the following problems.

a) Sodium and chlorine combine in an approximate ratio of 23 : 36 by mass to make table salt. Approximately how many grams of chlorine are there in 1 kg of table salt?

b) Fool's gold is a combination of iron and sulphur in an approximate ratio of 56 : 64 by mass. Approximately how many grams of sulphur are there in 1 kg of fool's gold?

4. Solve the following problems.

a) What fraction of 3 tons is 750 lb (pounds)? 1 ton = 2 000 lb.

b) If 7.5 feet of uniform wire weighs 10 pounds, what is the weight of 2 yards of the same wire in kilograms? 1 lb = 453.59 g, 1 yd = 3 ft.

c) If a cubic inch of metal weighs 2 pounds, a cubic foot of the same metal weighs how many kilograms? 1 ft = 12 in.

5. In a quadrilateral ABCD, AB = CB and AD = CD.

1) Prove that Angle (A) = Angle (C).

2) Prove that any point on the diagonal BD is equidistant from A and C.

3) Prove that AC _|_ BD.

6. Solve the following problems.

(a) The stage crew needed to place a row of stools along the counter of a restaurant. The stools were 1 ¹/₂ feet wide. How many stools were placed along the counter, which was 13 ⁵/₈ feet long?

(b) The diameter of a coin is ³/₄ inches. Coins are placed on a rectangular sheet measuring 6 inches by 12 inches. If no coins overlap or overhang the edge of the sheet, what is the largest number of coins that may be arranged on the sheet?

7. Suppose this pattern were continued:

a) How many toothpicks would be needed to make 9 triangles?

b) How would the number of toothpicks be fond if the number n of triangles were known? Write an expression for the number of toothpicks in terms of the number n of triangles.

8. Make the statements true by inserting grouping symbols (parenthesis) and any of the four operations (+, -, x, :).

1)   7   7   7   7   =   ²/₇

2)   9   9   9   9   =   8 ⁸/₉

9. Using a 9 inch x 12 inch sheet of paper lengthwise, a typist leaves a 1-inch margin on each side and a 1.5 inch margin on top and bottom. What fractional part of the page is used for typing?

10. Find the value of the following (do without a calculator and show all your work).

11. Find the missing digits in the problem below.

12. Find the number of digits that are required to number the pages of a book from 1 to 250.

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Answers 1. aaa 1) ¹/₂; aaaaaaaaaaaa 2) ¹/₄.

2. aa a) $112.70; aaaaaaaa 2) $42.75.

3. aaa a) 610 g; aaaaaaaaa b) 533 g.

4. aa a) ¹/₈; aaaaa b) 3.629 kg; aaaaa c)1 567.6 kg.

5.

a # a	a Statement	a Reason
a 1. aaa	a AB = CB.	a Given
a 2. aaa	a AD = CD.	a Given
a 3. a	a The triangles DAB and DCB are congruent.	a SSS: statements 1 and 2, DB is common.
a 4. aaa	a Angle (DAB) = Angle (DCB).	a Statement 3.
a 5. aaa	a Angle (ADB) = Angle (CDB).	a Statement 3.
a 6. aaa	a F is a point on the diagonal BD.	a Given
a 7. a	a The triangles AFD and CFD are congruent.	a SAS: statements 1, 5 and 6, DF is common.
a 8. aaa	a AF = CF.	a Statement 7.
a 9. a	a The triangles AED and CED are congruent, where E is the point of intersection of the diagonals AC and BD.	a Statements 6 and 7.
a 10. aaa	a Angle (AED) = Angle (CED) = 90o.	a Statement 9, supplementary angles.
a 11. aaa	a AC _|_ BD	a Statement 10.

6. aaa a) 9; aaaaaa b) 128.

7. aaa 19, 2n + 1.

8.

1) a (7 + 7) : 7 : 7 a = a ²/₇;

2) a 9 - 9 : 9 : 9 a = a 8 ⁸/₉.

9. aa ⁵/₉.

10. aa 1) 145; aaa 2) 125.

11.

12. a 642.

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Lesson 09

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1. Calculate mentally and record your answer.

2. Solve the following problems.

a) Jeremy bought a calculator for $29.95 and some computer paper for $24.50 at the same store. What did Jeremy pay for calculator and paper, including GST and PST if the GST is 7% and PST is 8%?

b) The cost of a sweater, including 7% sales tax, is $42.75. What is the price of the sweater?

3. Solve the following problems.

a) Jacques is given two pieces of string, each 12 cm long. With one piece he forms a rectangle whose length is double its width. With the other piece he forms a square. Find the ratio of the area of the square to the area of the rectangle.

b) Two positive numbers are in the ratio 5 : 9. If the sum of their squares is 424, find the numbers.

4. Solve the following problems.

(a) How many gallons of water are needed to feel a fish tank that measures 11" high, 14" long, and 9" wide? 1 gal = 3.785 L, 1 in = 2.54 cm.

(b) The water level of a swimming pool, 75 feet by 42 feet, is to be raised four inches. How many gallons of water must be added to accomplish this? 1 ft = 12 in.

5. Prove that

1) any radius of a circle which bisects a chord is perpendicular to the cord.

2) any radius of a circle which is perpendicular to a cord bisects the chord.

6. Solve the following problems.

(a) At NJL High School, ¹/₄ of the school's population are seniors, ¹/₅ are juniors and ¹/₃ are sophomores. If there are 390 freshmen, what is the total school population?

(b) If one-third of the liquid contents of a can evaporate on the first day and three-fourths of the remainder evaporates on the second day, what is the fractional part of the original contents remaining at the close of the second day?

7. Suppose this pattern were continued:

a) How many toothpicks would be needed to make 13 squares?

b) How would the number of toothpicks be fond if the number n of squares were known? Write an expression for the number of toothpicks in terms of the number n of squares.

8. Make the statements true by inserting grouping symbols (parenthesis) and any of the four operations (+, -, x, :).

1)   4   5   2   1   =   ³/₇

2)   6   2   2   2   =   ¹/₃₇

9. What is the area of an isosceles right-angles triangle with a hipotenuse 6 cm long?

10. Find the value of the following (do without a calculator and show all your work).

11. Each letter represents a different digit. Find the values of each letter.

12. The number of digits used to number all the pages of a book was 570. Find the number of pages in the book.

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Answers 1. aa 1) 3 ¹/₂; aaaaaaaaa 2) 9.

2. aa a) $62.62; aaaaaaa 2) $39.95.

3. aa a) 9 : 8; aaaaaaaaa b) 10 and 18.

4. aa a) 6 gallons; aaaaa b) 7 855.4 gallons.

5.

a)

a # a	a Statement	a Reason
a 1. aaa	a AC = BC.	a Given
a 2. aaa	a OA = OB.	a Radii of the circle with center O.
a 3. a	a The triangles AOC and BOC are congruent.	a SSS: statements 1 and 2, OC is common.
a 4. aaa	a Angle (ACO) = Angle (BCO) = 90o.	a Statement 3, supplementary angles.
a 5. aaa	a OC _|_ AB	a Statement 4.

b)

a # a	a Statement	a Reason
a 1. aaa	a OD _|_ AB	a Given.
a 2. aaa	a Angle (ACO) = Angle (BCO) = 90o.	a Statement 1.
a 3. aaa	a OA = OB.	a Radii of the circle with center O.
a 4. a	a The triangles AOC and BOC are congruent.	a HST: statements 2 and 3, OC is common.
a 1. aaa	a AC = BC.	a Given

6. aaa a) 1800; aaaaaa b) ¹/₆.

7. aaa 40, 3n + 1.

8.

1) a (4 + 5) : 2 1 a = a ³/₇;

2) a 6 : 2 2 2 a = a ¹/₃₇.

9. aa 9 cm².

10. aa 1) 5.003; aaa 2) 100.

11. aa A = 9, B = 2, C = 1.

12. a 226.

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Lesson 10

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1. Place any of the four operations (+, -, x, :) in the empty boxes to make each statement true.

2. Solve the following problems.

a) The cost of a new automobile, including 7% sales tax, is $18 190. What is the price of the automobile?

b) A record album, which regularly sells for $14.75, is on the sale at a discount of 20%. How much would you pay for the record, including the sales tax of 15%?

3. Solve the following problems.

a) Two positive integers are in the ratio 6 : 7. If the product of two integers is 6 048, find the larger integer.

b) The angles of a triangle are in the ratio 3 : 4 : 5. What are their measures?

4. Solve the following problems.

(a) Sometimes the apple cider is delivered in barrels. 1 barrel = 119.23 L. Tomorrow, Alex and his friends have to empty 22 barrels of apple cider into liter containers. How many containers will they need?

(b) A rectangular block of metal weighs 3 ounces (oz). How many kilogramss will a similar block of the same metal weigh if the edges are twice as large? 1 lb = 16 oz = 453.59 g

5. In the diagram, AB = DC and AC = DB. Prove that AE = DE.

6. Solve the following problems.

(a) Chris ate ³/₅ of the cookies that Dale baked. If Chris ate 15 cookies, find the number of cookies left for Dale.

(b) Each year a car depreciates to ⁴/₅ of the value one year before. What was the original value of a car that is worth $8 000 after 4 years?

7. A series of cubes are placed together as shown. The total number of faces that show are counted.

Suppose this pattern were continued.

a) How many faces would show on the 15-th diagram?

b) If the number of cubes were known, how could the number of faces be found? Write an expression for the number of faces in terms of the number n of cubes.

8. Make the statements true by inserting grouping symbols (parenthesis) and any of the four operations (+, -, x, :).

1)   3   4   6   7   =   ¹/₆

2)   4   3   2   1   =   ¹/₃

9. In a triangle ABC, AB = 7 cm, D is a point on BC such that BD = AD. Find the perimeter of the triangle ABC, if the perimeter of the triangle ACD is 15 cm.

10. Find the value of the following (do without a calculator and show all your work).

11. Each letter represents a different digit. Find the values of each letter.

12. Each of the numbers 1, 2, 3, and 4 is represented by one of the letters A, B, C, and D (not necessarily in that order). Find the largest possible sum of the 3-digit numbers BAC, CAD, and DAC.

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Answers 1. aa 1) : aaaaa 2) x aaaaa 3) - aaaaa 4) - aaaaa

2. aa a) $17 000; aaaaaaa 2) $13.57.

3. aa a) 72, 84; aaaaaaaaa b) 45^o, 60^o, and 75^o.

4. aa a) 2 623; aaaaa b) 0.68 kg.

5.

a # a	a Statement	a Reason
a 1. aaa	a AB = DC.	a Given
a 2. aaa	a AC = DB.	a Given.
a 3. a	a The triangles ABD and DBC are congruent.	a SSS: statements 1 and 2, AD = DA.
a 4. aaa	a Angle (EAD) = Angle (EDA) = 90o.	a Statement 3.
a 5. aaa	a AE = DE.	a Statement 4, ITT, part 2.

6. aaa a) 10; aaaaaa b) $ 19 531.25.

7. aaa 79, 5n + 4.

8.

1) a (3 + 4) : 6 : 7 a = a ¹/₇;

2) a (4 - 3) : (2 + 1) a = a ¹/₃.

9. aa 22 cm.

10. aa 1) 1.72; aaa 2) 6.875.

11. aa 1) A = 1, B = 5, C = 4, D = 2, E = 0; aa 2) A = 2, B = 3, C = 5, D = 9

12. a 941.

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Appendix 1

Triangle's Classification.

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Triangles may be classified by measures of their angles.

Triangle	Description	Example
acute triangle	all angles are acute
right triangle	one angle is 90o
obtuse triangle	one angle is obtuse

Triangles are also classified according to the lengths of their sides.

Triangle	Description	Example
equilateral triangle	3 sides equal
isosceles triangle	at least 2 sides equal
scalene triangle	no sides equal

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Appendix 2

Congruent Triangles. Isosceles Triangle.

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If two geometric figures have the same size and shape, they are said to be congruent. Two congruent geometric figures can made to coincide through a physical motion. Congruent triangles are identical in all respects. Their corresponding sides and corresponding angles are equal. For this reason, knowing that two triangles are congruent helps us to prove theorems relating corresponding angles or corresponding sides. The order in which the vertices are listed when pairs of triangles are congruent indicates which angle and side correspond.

However, it is not necessary to know this much information to show that two triangles are congruent. The following theorems give conditions that are sufficient to show that two triangles are congruent.

Side - Angle - Side Theorem (SAS)

If two sides and the contained angle of one triangle are respectively equal to the two sides and the contained angle of another triangle, then the triangles are congruent.

Given: Triangles ABC and A'B'C' such that AB = A'B', AC = A'C', and Angle (A) = Angle (A').

Required to Prove: The triangles ABC and A'B'C' are congruent.

Analysis: To prove that triangles ABC and A'B'C' are congruent, we need to show that we can map the triangle ABC onto the triangle A'B'C' so that A maps onto A', B maps onto B', and C maps onto C'.

Proof:

#	Statement	Reason
1.	AB = A'B'.	Given
2.	AC = A'C'.	Given
3.	Angle (A) = Angle (A').	Given
4.	We can move the triangle ABC onto the triangle A'B'C' so that A maps onto A',   AB falls along A'B', and AC falls along A'C'.	Statement 3.
5.	B maps onto B'.	Statement 1.
6.	C maps onto C'.	Statement 2.
7.	The motion discribed in the statement 4 maps the triangle ABC onto the triangle A'B'C'    so that A maps onto A', B maps onto B', and C maps onto C'.	Statements 4, 5, and 6.
8.	The triangles ABC and A'B'C' are congruent.	Statement 7.

Angle - Side - Angle Theorem (ASA)

If two angles and the contained side of one triangle are respectively equal to the two angles and the contained side of another triangle, then the triangles are congruent.

Given: Triangles ABC and A'B'C' such that Angle (B) = Angle (B'), Angle (C) = Angle (C'), and BC = B'C'.

Required to Prove: The triangles ABC and A'B'C' are congruent.

Proof:

#	Statement	Reason
1.	Angle (B) = Angle (B').	Given
2.	Angle (C) = Angle (C').	Given
3.	BC = B'C'.	Given
4.	We can move the triangle ABC onto the triangle A'B'C' so that B maps onto B',   BC falls along B'C', and BA falls along B'A'.	Statement 1.
5.	C maps onto C'.	Statement 3.
6.	CA falls along C'A'.	Statement 2.
7.	A lies on B'A'.	Statement 4.
8.	A lies on C'A'.	Statement 6.
9.	A maps onto A'.	Statements 7 and 8.
10.	The motion discribed in the statement 4 maps the triangle ABC onto the triangle A'B'C'    so that A maps onto A', B maps onto B', and C maps onto C'.	Statements 4, 5, and 9.
11.	The triangles ABC and A'B'C' are congruent.	Statement 10.

Isosceles Triangle

Glossary

1. Isosceles triangle. A triangle with two sides equal.

2. Equilateral triangle. A triangle with all sides equal.

3. Altitude or height of a triangle. A perpendicular line segment from a vertex to the side opposite that vertex.

4. Median of a triangle. A line segment joining he vertex to the midpoint of the opposite side.

5. Angle bisector. A line that divides an angler into two equal parts.

Isosceles Triangle Theorem. (ITT) Part 1

In an isosceles triangle:

a) the angles opposite the equal sides are equal;

b) the bisector of the angle formed by two equal sides is also an altitude and a median of the triangle.

Given: Isosceles triangle ABC with AB = AC, AD is the bisector of the angle BAC.

Required to Prove: Angle (B) = Angle (C), AD is perpendicular to BC, BD = CD.

Proof:

#	Statement	Reason
1.	AB = AC.	Given
2.	Angle (BAD) = Angle (CAD).	Given: AD bisects angle BAC.
3.	Triangles BAD and CAD are congruent.	SAS: statements 1 and 2, AD is common.
4.	Angle (B) = Angle (C).	Statement 3.
5.	BD = CD.	Statement 3.
6.	Angle (ADB) = Angle (ADC).	Statement 3.
7.	Angle (ADB) + Angle (ADC) = 180o.	Supplementary angles.
8.	Angle (ADB) = Angle (ADC)= 90o.	Statements 6 and 7.
9.	AD is perpendicular to BC.	Statement 8.

Corollary. In an equilateral triangle, the angles are equal.

Isosceles Triangle Theorem (ITT). Part 2

If a triangle has two equal angles, then the sides opposite the equal angles are equal.

Given: Triangle ABC such that Angle (B) = Angle (C).

Required to Prove: AB = AC.

Proof:

#	Statement	Reason
1.	Angle (ABC) = Angle (ACB).	Given
2.	BC = CB.
3.	Triangles ABC and ACB are congruent.	ASA: statements 1 and 2.
1.	AB = AC.	Statement 3.

Side - Side - Side Theorem (SSS)

If three sides of one triangle are respectively equal to the three sides of another triangle, then the triangles are congruent.

Given: Triangles ABC and A'B'C' such that AB = A'B', AC = A'C', and BC = B'C'.

Required to Prove: The triangles ABC and A'B'C' are congruent.

Proof:

#	Statement	Reason
1.	AB = A'B'.	Given
2.	AC = A'C'.	Given
3.	BC = B'C'.	Given
4.	We can move the triangle ABC onto the triangle A'B'C' so that B falls on B',   BC falls along B'C', and A falls on A" on the side of B'C' remote from A' Join A'A".
5.	C maps onto C'.	Statement 3.
6.	The triangle ABC maps onto the triangle A"B'C'.	Statements 4 and 5.
7.	The triangles ABC and A"B'C' are congruent.	Statement 6.
8.	AB = A"B'.	Statement 7.
9.	AC = A"C'.	Statements 7.
10.	A'B' = A"B'.	Statements 1 and 8.
11.	A'C' = A"C'.	Statements 2 and 9.
12.	Angle (B'A'A") = Angle (B'A"A').	ITT, statement 10.
13.	Angle (C'A'A") = Angle (C'A"A').	ITT, statement 11.
14.	Angle (B'A'C') = Angle (B'A"C').	Statements 12 and 13.
15.	The triangles A'B'C' and A"B'C' are congruent.	SAS, statements 10, 11, and 14.
16.	The triangles ABC and A'B'C' are congruent.	Statements 7 and 15.

Angle - Angle - Side Theorem (AAS)

If two angles and a side of one triangle are respectively equal to two angles and a side of another triangle, then the triangles are congruent.

Given: Triangles ABC and A'B'C' such that Angle (A) = Angle (A'), Angle (B) = Angle (B'), and BC = B'C'.

Required to Prove: The triangles ABC and A'B'C' are congruent.

Proof:

#	Statement	Reason
1.	Angle (A) = Angle (A').	Given
2.	Angle (B) = Angle (B').	Given
3.	BC = B'C'.	Given
4.	Angle (C) = Angle (C').	Sum of the Angles Theorem: the sum of the measures of the interior angles of a triangle is 180o.
5.	The triangles ABC and A'B'C' are congruent.	ASA, statements 2, 3, and 4.

Hypotenuse - Side Theorem (HST)

In a right triangle the side opposite to the right angle is called the hypotenuse. The other two sides are called legs.

If the hypotenuse and one leg side of one triangle are respectively equal to the hypotenuse and one leg of another triangle, then the triangles are congruent.

Given: Triangles ABC and A'B'C' such that Angle (C) = Angle (C') = 90^o, AB = A'B', and AC = A'C'.

Required to Prove: The triangles ABC and A'B'C' are congruent.

Proof:

#	Statement	Reason
1.	Angle (C) = Angle (C') = 90o.	Given
2.	AB = A'B'.	Given
3.	AC = A'C'.	Given
4.	Move the triangle ABC onto the triangle A'B'C' so that A falls on A',   AC falls along A'C', And B falls on B" on the side of A'C' remote frome B'.
5.	C maps onto C'.	Statement 3.
6.	The triangle ABC maps onto the triangle A'B"C'.	Statements 4 and 5.
7.	The triangles ABC and A'B"C' are congruent.	Statement 6.
8.	AB = A'B".	Statement 6.
9.	Angle (A'C'B") = Angle (A'C'B') = 90o.	Statement 6.
10.	Angle (B'C'B") is a straight angle.	Statement 9.
11.	B', C', and B" are collinear points.	Statement 10.
12.	A'B' = A'B".	Statements 2 and 8.
13.	Angle (A'B'C') = Angle(A'B"C').	Statements 11, 12, ITT.
14.	The triangles ABC and A'B"C' are congruent.	AAS, statements 9, 12 and 13.
15.	The triangles ABC and A'B'C' are congruent.	Statements 7 and 14.

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Appendix 3

Conditional Statements. Perpendicular and Angle Bisectors.

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Definition. A statement which begins with an "if" clause (or some equivalent) and ends with a "then" clause (or some equivalent) is called a conditional statement. The "if" clause is called "hypothesis" and the "then" clause is called the "conclusion".

Definition. The converse is the statement obtained by interchanging the hypothesis and the conclution.

#	Statement	Converse
1.	If the sum of the digits of a number is divisible by 3 then the number is divisible by 3.	If a number is divisible by 3 then the sum of its digits is divisible by 3.
2.	Every rectangle is a parallelogram.	Every parallelogram is a rectangle (false).

Clearly, the converse of a true statement is not necessarily true. We observe in the table above that the converse of the second statement is false while the first statement and its converse are true. When a statement and its converse are both true, we can combine them into a single biconditional statement using the "if and only if" construction.

A number is divisible by 3 if and only if the sum of its digits is divisible by 3.

The "if and only if" construction is used so often that mathematicians have created the word "iff" to read "if and only if". Therefore, this statement and its converse can be written as follows.

A number is divisible by 3 iff the sum of its digits is divisible by 3.

Perpendicular Bisector of a Line Segment

Definition. A line passes through the midpoint of a given line segment AB and perpendicular to this segment is called the perpendicular bisector of AB.

Theorem. If a point is on the perpendicular bisector of a line segment, then it is equidistant from the ends of the line segment.

Given: P is a point on the perpendicular bisector of AB, M is a midpoint of AB.

Required to Prove: PA = PB.

Proof.

#	Statement	Reason
1.	AM = BM.	PM is the perpendicular bisector.
2.	PM _|_ AB.	PM is the perpendicular bisector.
3.	Angle (AMP) = Angle (BMP) = 90o.	Statement 2.
4.	The triangles AMP and BMP are congruent.	SAS, statements 1 and 3, PM is common.
5.	PA = PB.	Statement 4.

Now, we prove the converse of the theorem above.

Theorem. If a point is equidistant from two given points, then it is on the perpendicular bisector of the line segment joining the two points.

Given: AB is a line segment, P is a point such that PA = PB.

Required to Prove: P is on the perpendicular bisector of AB.

Plan. Let M be the midpoint of AB. Join PM. Using congruent triangles prove that Angle (AMP) = Angle (BMP) = 90^o.

Proof:

#	Statement	Reason
1.	PA = PB.	Given.
2.	AM = BM.	By construction.
3.	The triangles AMP and BMP are congruent.	SAS, statements 1 and 2, PM is common.
4.	Angle (PMA) = Angle (PMB).	Statement 3.
5.	Angle (PMA) + Angle (PMB) = 180o.	Supplementary angles.
6.	Angle (PMA) = Angle (PMB) = 90o.	Statements 4 and 5.
7.	PM _|_ AB.	Statement 6.
8.	PM is a perpendicular bisector of AB.	Statements 2 and 7.

The two theorems proved above can be combined in a single biconditional statement called the Perpendicular Bisector Theorem. When a statement and its converse are both true, we can combine them into a single biconditional statement using the "if and only if" construction, This construction is used so often that mathematicians have created the word "iff" to read "if and only if".

Perpendicular Bisector Theorem.

A point is on the perpendicular bisector of a line segment iff it is equidistant from the ends of the line segment.

Angle Bisector

A ray that divides an angle into two angles having the same measure and has the vertex of the angle as the endpoints is called the angle bisector.

Theorem. If a point is on the bisector of an angle, then it is equidistant from the arms of the angle.

Given. P is a point on the bisector of the Angle (ABC).

Required to Prove: P is equidistant from AB and AC.

Analysis. Drop the perpendiculars from P onto AB and AC at E and D respectively. We are required to prove that PE = PD.

Proof.

#	Statement	Reason
1.	Angle (PAE) = Angle (PAD).	Given.
2.	Angle (PEA) = Angle (PDA) = 90o.	By construction.
3.	The triangles PEA and PDA are congruent.	AAS, statements 1 and 2, AP is common.
4.	PE = PD.	Statement 3.

Here is the converse theorem.

Theorem. If a point is equidistant from the arms of an angle, then it is on the bisector of the angle.

Given: PE and PD are the perpendiculars from a point P to the arms AB and AC of the angle BAC such that PE = PD.

Required to Prove: Angle (PAE) = Angle (PAD).

Proof.

#	Statement	Reason
1.	PE _|_ AB.	Given.
2.	PE = PD.	Given.
3.	Angle (PEA) = Angle (PDA) = 90o.	By construction.
4.	The triangles PEA and PDA are congruent.	HS, statements 2 and 3, AP is common.
5.	Angle (PAE) = Angle (PAD).	Statement 4.
6.	AP is the bisector of Angle (BAC).	Statement 5.

This theorem and its converse can be combined into the biconditional statement called the Angle Bisector Theorem.

Angle Bisector Theorem

A point is on the bisector of an angle iff it is equidistanat from the arms of the angle.

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