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Mathematics 20 Module 3 Lesson 20 Review of Lessons 14-19 Mathematics 20 329 Lesson 20 Mathematics 20 330 Lesson 20 Review of Lessons 14-19 Introduction This lesson will be a review of the concepts that you have learned in Lessons 14-19. Lessons 14 and 15 were the consumer math section where you have seen the important link of mathematics to facing the financial challenges of everyday life. In lessons 16 to 19 you have used your mathematical skills to explore the world of geometry. Each section will be set up so that it is a review of a lesson. A number of practice exercises will be provided with each review. Work through the practice exercises so that you feel comfortable with the concepts in that particular lesson. If you find that you are having difficulties, go back to the actual lesson and review the material there. It is so important to understand the material and be able to apply it to a number of different situations. The assignment for this lesson will be a review of the entire course. It will be very similar to the final examination that you will receive at the conclusion of this course. Every aspect of the course will be covered. Once again, if there is some area that you are having problems with, go back to those sections and review the material. Mathematics 20 331 Lesson 20 Mathematics 20 332 Lesson 20 Objectives After completing this lesson you will be able to • solve problems involving the mathematical concepts in Lessons 14 to 19. • apply the mathematical concepts from Lessons 14 to 19 to solve real-world problems. Mathematics 20 333 Lesson 20 Mathematics 20 334 Lesson 20 20.1 Review of Lesson 14 Some important concepts to remember in this section are: • there are many advantages and disadvantages of credit. Be informed! • the Simple Interest Formula is: I = prt where: • • • • I is the interest. p is the principle amount. r is the interest rate. t is the time in number of days. • Interest on purchases is charged only if the balance owing is not paid in full by the due date. If the balance is not paid in full, interest will be charged on the unpaid balance and on any purchases that were made during the month, up to and including the date of the billing statement. • the Monthly Payment Formula is: Monthly Payment Formula r 12 M=p 1 1 12 t r 1 + 12 Where: • • • • Mathematics 20 M is the amount of the monthly payment. p is the principal or the amount of the loan. r is the annual interest rate. 9 = 0 .09 = r Example: 9 % = 100 r is the monthly interest rate 12 t is the term in years. 12t is the term in months 335 Lesson 20 Exercise 20.1 1. Calculate the cost of credit of each of these items. Item Cash Price Monthly Payment Number of Months a) Mountain Bike $425.90 $51.80 10 b) CD Player $215.33 $28.48 8 c) Golf Clubs $639.85 $66.43 12 d) Cordless Telephone $148.50 $40.00 4 e) Gold Bracelet $234.99 $42.99 6 f) 2. Which item has the lowest percentage increase in the final cost? Determine the cost of credit for each option and decide which option is the best. Cash Price Option A Option B a) $1490.55 $260.00 per month for 6 months $156.50 per month for 12 months b) $883.67 $101.00 per month for 9 months $300.00 per month for 3 months c) $540.99 $70.59 per month for 10 months $100.00 per month for 6 months d) $5189.46 $250.00 per month $200.00 per month for 2 years for 3 years e) Mathematics 20 What can you say, in general, about the options for paying back a loan? 336 Lesson 20 3. 4. Calculate the amount of interest that has accrued by the statement date which is the first of the month, and determine the balance that will be carried forward to the next statement. Balance on Last Statement Payment New Purchases Cash Advances Daily Interest Rate Due Date a) $130.80 $25.00 (June 3) $61.99 (June 13) - 0.05% July 1 b) $644.07 $125.00 (Dec. 23) $35.82 (Dec. 7) $100.93 (Dec. 22) - 0.055% Jan. 1 c) $431.00 $100.00 (Jan. 24) $452.01 (Jan.10) $50.00 (Jan.19) 0.04% Feb. 1 d) $98.62 $60.00 (Mar. 28) $29.99 (Mar. 6) $54.00 (Mar. 14) $60.00 (Mar. 11) $40.00 (Mar. 20) 0.045% Apr. 1 Vern wants to buy a boat that costs $7120.00. The interest rate at the bank is 1 10 %. Determine the monthly payments if Vern borrows the money over 4 years. 2 How much would Vern save if he borrowed the money over 2 years? Mathematics 20 337 Lesson 20 5. Cindy and Lin are buying a new home. The amount of the mortgage will be $81 500 and will be paid over 20 years at an interest rate of 7%. What will their monthly payments be? 6. Sajamin borrows $6350.00 to buy a used car. The interest rate is 12% and Sajamin will pay for the car over 4 years. What are monthly payments? 7. Paul's monthly payments on a new stereo system are $130.00 each month. When Paul bought the system, the interest rate was 9% and the length of the term was 3 years. What was the original cost of the stereo system? Mathematics 20 338 Lesson 20 20.2 Review of Lesson 15 `` Some important concepts to remember in this section are: • the final cost of a taxable item is: Final Cost = Original Cost + 6% PST x Original Cost + 5% GST x Original Cost • The formula for mill rate is: Mill Rate = • Budgeted Expenses Budgeted Expenses = 1000 Assessment Assessment 1000 The formula for property tax is: Assessment Property Tax = Mill Rate 1000 Exercise 20.2 1. Fill in the following table. Item Cost Coat $76.22 Hockey Skates GST PST Final Cost $14.31 Snowmobile $671.16 Calculator $63.50 2. The town of Cupar has a total budget of $187 971 assessed value of $2 725 220. What is its municipal mill rate? 3. In Cupar, the Howards own a home that is worth $8 620 and the property is worth $1 750. What are their municipal property taxes? Mathematics 20 339 Lesson 20 4. The Howards taxes are due June 30. The discount that the Howards receive for paying their taxes is listed in the table. a) b) March 30 April 30 May 30 June 30 5% 4% 2% 0% What are their municipal taxes if they pay by March? What are their municipal taxes if they pay by May? 5. Sid and Mary Lou own a farm in the R.M. of Sherwood. Their taxable assessment is $12 860. The municipal tax is 24.00 mills. The School tax is 78.8 mills. How much will Sid and Mary Lou pay in taxes? Separate the amount for municipal and school taxes and then find the total. 6. Sid and Mary Lou’s taxes are due on Dec. 30, but they plan to pay their taxes by September 30th. According to the table for discounts in question #4, how much would they pay? What would be the amount of money that they would save? (March 30 is the same as September 30) 7. A penalty of 0.75 % per month will be added to taxes that are not paid on time. How much would Sid and Mary Lou pay if they didn't pay until May of the following year? 8. Calculate the property taxes. In each of the following cases the mill rate is given for that particular municipality. Take into consideration the date the taxes would be paid and apply the discount or penalty according to the table in question 4. Taxes are due June 30 with a penalty of 0.75% per month late fee. Assessed Land Value Mill Rates Building Municipal Tax Library Rate Public School a) $6 000 $7 670 41.05 --- 69.46 February b) $13 000 $24 716 52.86 8.24 72.33 July c) $9 576 $10 265 53.10 --- 74.55 May d) $12 837 $31 930 48.62 8.89 79.02 September e) $4 126 $8 980 49.00 --- Mathematics 20 340 73.99 Separate School Payment Date March Lesson 20 20.3 Review of Lesson 16 Some important concepts to remember in this section are: • the word congruent means to have the same measure. • congruent segments are segments which have the same measure. • congruent angles are angles which have the same measure. • two triangles are congruent if the corresponding angles and sides of the two triangles are congruent. • the Congruence Postulates are used to determine if two triangles are congruent. • Side-Side-Side (SSS) Congruence Postulate If three sides of one triangle are congruent, respectively, to three sides of a second triangle, then the two triangles are congruent. • Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent, respectively, to two sides and the included angle of a second triangle, then the two triangles are congruent. • Angle-Side-Angle (ASA) Congruence Postulate If two angles and the included side of one triangle are congruent, respectively, to two angles and the included side of a second triangle, then the two triangles are congruent. • Angle-Angle-Side (AAS) Congruence Postulate If two angles and the non-included side of one triangle are congruent, respectively, to two angles and the non-included side of a second triangle, then the two triangles are congruent. • Hypotenuse-Leg (HL) Theorem If the hypotenuse and a leg of a right triangle are congruent, respectively, to the hypotenuse and a leg of a second right triangle, then the two right triangles are congruent. • triangles can be constructed through both informal and formal methods. The formal method of construction is done with a straight edge and compass. Mathematics 20 341 Lesson 20 Exercise 20.3 1. 2. For each of the pairs of triangles, like markings indicate the congruent parts. Name the congruence postulate, if any, that will prove the triangles congruent. a. b. c. d. e. f. g. h. i. j. k. l. List the corresponding parts of the two triangles if FUN DAY . Mathematics 20 342 Lesson 20 RST with SL TK and RST RTS . RSL RTK 3. Given: Prove: 4. Given the following three segments, construct FGH . What congruence postulate determines that you can do this? 5. Given the following two angles and segment construct WXY . What congruence postulate determines that you can do this? 20.4 Review of Lesson 17 Some important concepts to remember in this section are: • there are many different reasons to use when proving two segments or angles are congruent and these are listed in the appendix. • when proving two triangles are congruent, always mark the congruent parts on the two triangles first. Then determine which congruence postulate you are going to use. • the last statement in the two-column proof should be the same as what the question asked you to prove. • once you have proven that two triangles are congruent, you can also say that the corresponding parts are also congruent. • when constructing a right triangle it is necessary to construct a perpendicular line from one of the legs. Mathematics 20 343 Lesson 20 Exercise 20.4 1. Mark the congruent parts on the triangles from the information that is given. a. Given: Isosceles JKL with RJ bisecting KJL b. Given: Figure ABCD with AB AD , BC DC , AB CD c. Given: Figure with RS TU , V is the midpoint of ST . d. Given: ABC with AR BS , RSC is isosceles. Mathematics 20 344 Lesson 20 e. Given: X XYZ with altitudes YN to XZ ; ZM to XY and XM XN . M 1 2 Y f. 2. Given: 3 4 N Z AM is the bisector to BC . Using the information given in #1, prove the following. a. Prove: JKR JLR b. Prove: ADB CBD c. Prove: RSV UTV d. Prove: ACR BCS e. Prove: YN ZM f. Prove: ABC is isosceles Mathematics 20 345 Lesson 20 3. Construct a right triangle with leg lengths, AB and CD. 20.5 Review of Lesson 18 Some important concepts to remember in this section are: • two polygons are similar if their vertices can be paired so that: • corresponding angles are congruent. • corresponding sides are in proportion (lengths have the same ratio). • the scale factor is the ratio of the lengths of any two sides of the similar polygon. • if two solids are similar with a scale factor of a : b , then corresponding areas have a ratio of a 2 : b2 , and corresponding volumes have a ratio of a 3 : b3 . Exercise 20.5 1. Hexagon ABCDEF is similar to hexagon UVWXYZ. List the corresponding sides and vertices. Mathematics 20 346 Lesson 20 2. In each of the following pairs of polygons determine the lengths of the missing sides or the measures of the missing angles. a. b. c. d. 20 Mathematics 20 347 Lesson 20 3. Find the length of the missing side in the following similar right triangles. a. b. c. d. Mathematics 20 348 Lesson 20 4. 5. Determine the proportion of the area and volume for the following similar solid polygons. a) The length of the radius on cylinder A is 4 cm and on cylinder B is 8 cm. b) The width of box A is 9 cm and of box B is 12 cm. c) The height of pyramid B is 13.2 cm and pyramid A is 4.4 cm. From the ratios in question 4, determine the value of the i) area ii) volume of the solid B. a) Cylinder A • area • volume - 65.36 cm2 150.72 cm3 b) Box A • area • volume - 102 cm2 54 cm3 c) Pyramid A • area of base • volume - Mathematics 20 62 m2 294.36 m3 349 Lesson 20 20.6 Review of Lesson 19 Some important concepts to remember in this section are: • it is necessary to understand the vocabulary of circles. This was discussed in Section 19.1. • a central angle of a circle is an angle in the plane of the circle with its vertex coinciding with the centre of the circle. • a central angle forms both a major arc and a minor arc with the circle. • a circle can be both inscribed within a triangle and circumscribed around the triangle through the method of a formal construction. • a tangent can be constructed to a circle from a point outside the circle. • a number of theorems have been developed about circles. Exercise 20.6 1. Given circle with centre O and tangents AP and BP . a. If AP = 8 and OP = 10, what is the radius of the circle? b. If AP = 8 and OP = 10, then BP = _______________. c. If the measure of angle APB is 56°, how large is angle AOB? Mathematics 20 350 Lesson 20 2. 3. For the circle with centre O and CO perpendicular to AB, if the , then: , and a. m AC = i. m OCE = b. m CB = j. m CEO = c. m BD = k. m CFO = d. m AD = l. m BFE = e. m AE = m. m OFE = f. m COA = n. m AOD = g. m BOE = o. m OAD = h. m COE = In the circle with centre O chords AB and CD intersect at P. (See Assignment 19) a. If CP = 4, PD = 6, and PB = 3, then AP = __________. b. If BP = 4, AP = 15, and CP = 6, then PD = __________. Mathematics 20 351 Lesson 20 4. Chords AB and CD intersect at P. If CP = 4, PD = 10, PB = x, and AP = x + 3, find the value of x. Mathematics 20 352 Lesson 20 Answers to Exercises Exercise 20.1 1. a. b. c. d. e. f. $518.00 $227.84 $797.16 $160.00 $257.94 Golf clubs 2. a. Option A Option B $1 560.00 $1 878.00 b. Option A Option B $909.00 $900.00 Best c. Option A Option B $705.90 $600.00 Best d. Option A Option B $6 000.00 $7 200.00 e. Generally, less money will have to be paid back if the loan is paid over a shorter period of time. 3. a. b. c. d. $169.99 $667.30 $842.48 $225.52 4. $182.30/month for 4 years $330.20/month for 2 years. Best Best Savings of $825.60 if paid in 2 years instead of 4 years. Mathematics 20 5. $631.88 6. $167.22 7. $4 088.09 353 Lesson 20 Exercise 20.2 1. Item Cost GST PST Final Cost Coat $76.22 $4.57 $3.81 $84.60 Hockey Skates $238.50 $14.31 $4.93 $264.74 Snowmobile $13 423.20 $805.39 $671.16 $14 899.75 Calculator $57.21 $3.43 $2.86 $63.50 Exercise 20.3 Mathematics 20 2. 69 mills 3. $715.53 4. a. b. 5. School Tax - $1 013.37 Municipal Tax - $ 308.64 Total - $1 322.01 6. $1 255.91 $66.10 7. $1 371.59 8. a. b. c. d. e. $1 589.98 $5 070.19 $2 482.05 $6 249.56 $1 531.31 1. a. b. c. d. e. f. g. h. i. j. k. l. ASA SSS none none ASA SAS ASA none none none SSS none 679.75 701.22 354 Lesson 20 2. 3. Exercise 20.4 FU DA FN DY UN AY F D U A N Y 1. 2. 3. Statement SL TK RSL RTK R R 4. RSL RTK 4. Refer to 16.4 in lesson 16. SSS Postulate 5. Refer to 16.4 in lesson 16 ASA Postulate 1. a. Reason 1. Given 2. Given 3. Congruence is an equivalence relation on the set of triangles (reflexivity) 4. AAS Postulate b. Mathematics 20 355 Lesson 20 c. d. e. f. Mathematics 20 356 Lesson 20 2. a. 1. 2. Statement JKL is isosceles JK JL 3. 4. 5. 6. RJ bisects KJL KJR LJR JR JR JKR JLR Reason 1. Given 2. Definition of isosceles triangle 3. Given 4. Definition of bisect 5. Reflexive property 6. SAS Postulate 3. 4. 5. Statement AB AD , BC DC BAD and DCB are right angles AB CD BD DB ADB CBD Reason 1. Given 2. Perpendicular lines meet to form right angles 3. Given 4. Reflexive property 5. HL Theorem 1. 2. Statement RS TU S T 3. V is the midpoint of Reason 1. Given 2. If two lines are parallel, the alternate interior angles are congruent 3. Given 4. 5. ST SV TV SVR TVU 4. 5. 6. RSV UTV 6. b. 1. 2. c. Mathematics 20 357 Definition of midpoint Vertical angles are congruent. ASA Postulate Lesson 20 d. Statement AR BS RSC is isosceles CR CS 2 3 Reason 1. Given 2. Given 3. 4. Angles opposite 5. Angles that form a linear pair are supplementary 6. 1 & 2 are supplementary 3 & 4 are supplementary 1 4 6. 7. ACR BCS 7. Supplements of congruent angles are congruent SAS Postulate Statement Altitudes YN to XZ and ZM to XY YN XZ ; ZM YX XMZ & XNY are right angles XM XN X X XMZ XNY YN ZM Reason 1. Given Reason 1. Given 4. Statement AM is the bisector to BC AM BC AMB and AMC are right angles AMB AMC 5. 6. 7. 8. 9. BM CM AM AM AMB AMC AB AC ABC is isosceles 5. 6. 7. 8. 9. 1. 2. 3. 4. 5. e. 1. 2. 3. 4. 5. 6. 7. f. 1. 2. 3. Mathematics 20 358 2. 3. 4. 5. 6. 7. 2. 3. 4. Definition of altitude Perpendicular lines meet to form right angles Given Reflexive property ASA Postulate CPCTC Definition of bisector Perpendicular lines meet to form right angles Any two right angles are congruent. Definition of bisector Reflexive property SAS Postulate CPCTC Definition of isosceles Lesson 20 Exercise 20.5 3. Hint: Draw the perpendicular bisector to form a right angle. 1. A ~ U B ~ V X ~ W D ~ X E ~ Y F ~ Z 2. a. E = 58° F = 60° D = 62° A = 62° c = 7.5 d=6 b. x = 11.25 c. P 88 Q 140 N 123 w = 6.6 x = 11 y = 13.6 z = 14.6 d. y = 9.6 x = 7.2 a. b. c. d. x = 46.4 m x = 10.1 m x = 14.4 m x = 70.0 m 3. Mathematics 20 AB ~ UV BC ~ VW CD ~ WX DE ~ XY EF ~ YZ FA ~ ZU 359 Lesson 20 4. a. Area = 16 1 64 4 Volume = 1 8 b. Area = 9 16 Volume = 27 64 c. Area = 1 9 Volume = 1 27 5. Exercise 20.6 Mathematics 20 a. i. ii. Area B = 261.4 cm2 Volume B = 1205.76 cm3 b. i. ii. Area B = 181.3 cm2 Volume B = 128 cm3 c. i. ii. Area B = 558 m2 Volume B = 7947.72 m3 1. a. b. c. 6 8 124° (Hint: PAO PBO by the SSS Postulate) 2. a. b. c. d. e. f. g. h. 90° 90° 50° 130° 150° 90° 30° 120° 3. a. b. AP = 8 PD = 10 4. x=5 i. j. k. l. m. n. o. 360 30° 30° 60° 30° 120° 130° 25° Lesson 20 Mathematics 20 Module 3 Assignment 20 Mathematics 20 361 Lesson 20 Mathematics 20 362 Lesson 20 Optional insert: Assignment #20 frontal sheet here. Mathematics 20 363 Lesson 20 Mathematics 20 364 Lesson 20 Assignment 20 Values (40) A. Multiple Choice: Select the best answer for each of the following and place a check () beside it. 1. The value of 7 5 14 2 4 ____ ____ ____ ____ 2. 3. Mathematics 20 a. b. c. d. 3 is ***. 16 47 36 83 If x 25 x 3 y 3 x 2 the simplified equation with y isolated is ***. ____ a. ____ ____ b. c. ____ d. 3 5x 3 y 1 5x y 1 2x 1 6x y 3 y The graph of y x 2 is ***. ____ a. ____ c. ____ b. ____ d. 365 Lesson 20 4. The simplified form of ____ ____ ____ ____ 5. 6. ____ a. ____ b. ____ c. ____ d. 3 2 x> 1 x> 3 3 , x> 2 2 x < 1, x > 1 x< The equivalent form of ____ a. ____ b. ____ c. d. 2 is ***. 3 6 3 2 3 6 3 2 3 The expression x 2 9 is equivalent to ***. 2 ____ ____ ____ ____ Mathematics 20 62 6 2 2 104 42 30 32 The solution to x 3 x 1 5 is ***. ____ 7. a. b. c. d. 54 50 24 18 is ***. a. b. c. d. x 3 x 3 x 11 x 7 x 2 13 x 5 x 1 366 Lesson 20 8. When x 2 4 x 5 is divided by x + 1, the quotient is ***. ____ ____ ____ ____ 9. 10. 11. Mathematics 20 a. b. c. d. x 1 x 5 x 1 x 5 0 The equivalent simplifed form of ____ a. ____ ____ ____ b. c. d. a b0 4 a 4 2 a 2 b2 a bb 2 1 ab 1 2 ab The equivalent form of a 3 a 1 1 ____ a. ____ b. ____ c. ____ d. If f x ____ a. ____ b. ____ ____ c. d. is ***. 1 is ***. 4 a 2a 3 2 2 a 2a 3 a 3 a 1 2a 4 2 a 2a 3 2 2x 1 5 , then f(2) is ***. x 2 ( x 3) 7 7 5 0 not defined 367 Lesson 20 12. The interest for one month on a $800 amount at an interest rate of 8% is ***. ____ ____ ____ ____ 13. a. b. c. d. A D AB = DE AC DF AB DE a. b. c d. ABC FED AB EF BAC EDF B D From the diagram it is possible to conclude that ***. ____ ____ ____ ____ Mathematics 20 $251.00 $3 514.00 $140.00 $2 430.00 If ABC DEF , then it is also true that ***. ____ ____ ____ ____ 16. a. b. c. d. If ABC DEF , then because corresponding parts of congruent triangles are congruent, the immediate conclusion is ***. ____ ____ ____ ____ 15. $64.00 $8.00 $5.33 $5.21 The mill rate in a town is 251 mills. A home and lot assessed at $14 000 would have a property tax of ***. ____ ____ ____ ____ 14. a. b. c. d. a. b. c. d. ABC EDF ABC DEF B E CB EF 368 Lesson 20 17. If ABC is an equilateral then the measure of 1 is ***. ____ ____ ____ ____ 18. 19. ____ a. 3 ____ b. 3 ____ c. 4 ____ d. 4 1 2 1 2 If AB = 8 then AE is equal to ***. a. b. c. d. 2 3 4 6 A picture frame in the shape of a regular 10 sided polygon is to be constructed. At what angle should the frame pieces be cut? ____ ____ ____ ____ Mathematics 20 15 75 105 115 In the given diagram, the length of DC is equal to ***. ____ ____ ____ ____ 20. a. b. c. d. a. b. c. d. 36° 60° 72° 144° 369 Lesson 20 Mathematics 20 370 Lesson 20 Part B can be answered in the space provided. You also have the option to do the remaining questions in this assignment on separate lined paper. If you choose this option, please complete all of the questions on the separate paper. Evaluation of your solution to each problem will be based on the following. (40) B. • A correct mathematical method for solving the problem is shown. • The final answer is accurate and a check of the answer is shown where asked for by the question. • The solution is written in a style that is clear, logical, well organized, uses proper terms, and states a conclusion. 1. A rectangular box has sides whose lengths are x 2 cm, x 2 cm, and 2 x 1 cm. Write the polynomial in standard form which represents the volume of the box. What is the volume when x is 10 cm? Mathematics 20 371 Lesson 20 2. Calculate the length of the line segment from points 3, 2 to point 4, 1 on the coordinate plane. 3. Divide x 3 x 2 13 x 14 by x 2 . 4. Simplify Mathematics 20 x x 3 x 3 x 9 2 9 2 3 6 2 1 . 372 Lesson 20 5. 6. Mathematics 20 Simplify 2 5 2 . b 9 b 8 b 15 2 The velocity of an object at any time t in seconds is given by vt 3 t 2 47 t 30 Find the time at which the object is motionless. 373 Lesson 20 7. Solve the equation x 3 x 4 for all admissible roots. 8. Sketch the graph of y 3 x 5 3 and analyze it by discussing domain, range, x and y intercepts, axis of symmetry, vertex, concavity and maximum minimum points. Mathematics 20 2 374 Lesson 20 y x Mathematics 20 375 Lesson 20 Answer Part C on separate lined paper. Please include any tables or graphs that you are required to do with the assignment. (5) C. 1. Suppose that it is equally likely that a dart could land anywhere on the given figure. What is the probability that it will land in the circle? (10) (5) 2. 3. a. Given that DE is parallel to BC , prove that ADE ~ ABC . b. If BC = 2 find the length of DE. c. What is the ratio of the area of ADE to the area of ABC ? Prove APO BPO where A and B are points of tangency. Prove AOC BOC (5 mark Bonus) _____ (100) Mathematics 20 376 Lesson 20 Appendix Reasons for Proofs 1. Given 2. Definition of perpendicular lines • Two lines that meet or intersect to form right angles. 3. Definition of a right angle • An angle with a measure of 90 . 4. Definition of a bisector of an angle • The bisector of ABC is a ray BD in the interior of ABC such that ABD DBC . 5. Definition of the midpoint of a segment • The point that divides the segment into two congruent segments. 6. Definition of a segment bisector • A line, segment, ray, or plane that intersect the segment at its midpoint. 7. Definition of a perpendicular bisector of a segment • A line, ray, or segment that is perpendicular to the segment at its midpoint. 8. Definition of an altitude of a triangle • The segment from a vertex of a triangle and perpendicular to the line containing the opposite side. 9. Definition of a median of a triangle • A segment from a vertex of a triangle to the midpoint of the opposite side. 10. Definition of complementary angles • Two angles whose measures have the sum of 90 . Mathematics 20 377 Lesson 20 11. Definition of supplementary angles • Two angles whose measures have the sum of 180 . 12. Properties of Equality a) Addition Property If a b and c d , then a c b d . b) Subtraction Property If a b and c d , then a c b d . c) Multiplication Property If a b , then ca cb . d) Division Property If a b and c 0 , then a b . c c e) Reflexive Property aa f) Symmetric Property If a b , then b a . g) Transitive Property If a b and b c , then a c . h) Substitution Property If a b , then either a or b may be substituted for the other in any equation or inequality. 13. The sum of the measures of the angles of a triangle is 180 . 14. Vertically opposite angles are congruent. 15. Parallel Lines a) If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. b) If two parallel lines are cut by a transversal, then the corresponding angles are congruent. c) If two parallel lines are cut by a transversal, then the same side interior angles are supplementary. Mathematics 20 378 Lesson 20 16. Congruence Postulates a) b) c) d) e) f) SSS SAS ASA AAS HL LL Mathematics 20 379 Lesson 20 Mathematics 20 380 Lesson 20