4-1 Classifying Triangles (pp )

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1 Vocabulary acute triangle auxiliary line base base angle congruent polygons coordinate proof corollary corresponding angles corresponding sides CPCTC equiangular triangle equilateral triangle exterior exterior angle included angle included side interior interior angle For a complete list of the postulates and theorems in this chapter, see p. S82. isosceles triangle legs of an isosceles triangle obtuse triangle remote interior angle right triangle scalene triangle triangle rigidity vertex angle Complete the sentences below with vocabulary words from the list above. 1. A(n)? is a triangle with at least two congruent sides. 2. A name given to matching angles of congruent triangles is?. 3. A(n)? is the common side of two consecutive angles in a polygon. 4-1 Classifying Triangles (pp ) EXAMPLE Classify the triangle by its angle measures and side lengths. isosceles right triangle Classify each triangle by its angle measures and side lengths Angle Relationships in Triangles (pp ) EXAMPLE Find m S. 12x = 3x x 12x = 9x x = 42 x = 14 m S = 6 (14) = 84 Find m N In LMN, m L = 8x, m M = (2x + 1), and m N = (6x - 1). 284 Chapter 4 Triangle Congruence

2 4-3 Congruent Triangles (pp ) EXAMPLE Given: DEF JKL. Identify all pairs of congruent corresponding parts. Then find the value of x. The congruent pairs follow: D J, E K, F L, DE JK, EF KL, and DF JL. Since m E = m K, 90 = 8x After 22 is added to both sides, 112 = 8x. So x = 14. Given: PQR XYZ. Identify the congruent corresponding parts. 8. PR? 9. Y? Given: ABC CDA Find each value. 10. x 11. CD 4-4 Triangle Congruence: SSS and SAS (pp ) Given: RS UT, and VS VT. V is the midpoint of RU. Prove: RSV UTV Proof: Statements 1. RS UT 2. VS VT 3. V is the mdpt. of RU. 4. RV UV 5. RSV UTV Reasons 1. Given 2. Given 3. Given 4. Def. of mdpt. 5. SSS Steps 1, 2, 4 Show that ADB CDB when s = 5. AB = s 2-4s AD = 14-2s = (5) = 14-2 (5) = 5 = 4 BD BD by the Reflexive Property. AD CD and AB CB. So ADB CDB by SSS. 12. Given: AB DE, DB AE Prove: ADB DAE 13. Given: GJ bisects FH, and FH bisects GJ. Prove: FGK HJK 14. Show that ABC XYZ when x = Show that LMN PQR when y = 25. Study Guide: Review 285

3 4-5 Triangle Congruence: ASA, AAS, and HL (pp ) Given: B is the midpoint of AE. A E, ABC EBD Prove: ABC EBD 16. Given: C is the midpoint of AG. HA GB Prove: HAC BGC Proof: 1. A E Statements 2. ABC EBD 3. B is the mdpt. of AE. 4. AB EB 5. ABC EBD Reasons 1. Given 2. Given 3. Given 4. Def. of mdpt. 5. ASA Steps 1, 4, Given: WX XZ, YZ ZX, WZ YX Prove: WZX YXZ 18. Given: S and V are right angles. RT = UW. m T = m W Prove: RST UVW 4-6 Triangle Congruence: CPCTC (pp ) Given: JL and HK bisect each other. Prove: JHG LKG Proof: Statements 1. JL and HK bisect each other. 2. JG LG, and HG KG. 3. JGH LGK 4. JHG LKG 5. JHG LKG 1. Given Reasons 2. Def. of bisect 3. Vert. Thm. 4. SAS Steps 2, 3 5. CPCTC 19. Given: M is the midpoint of BD. BC DC Prove: Given: PQ RQ, PS RS Prove: QS bisects PQR. 21. Given: H is the midpoint of GJ. L is the midpoint of MK. GM KJ, GJ KM, G K Prove: GMH KJL 286 Chapter 4 Triangle Congruence

4 4-7 Introduction to Coordinate Proof (pp ) Given: B is a right angle in isosceles right ABC. E is the midpoint of AB. D is the midpoint of CB. AB CB Prove: CE AD Proof: Use the coordinates A(0, 2a), B(0, 0), and C (2a, 0). Draw AD and CE. By the Midpoint Formula, E = ( _ , _ 2a ) = (0, a) and D = ( _ 0 + 2a,_ ) = (a, 0) By the Distance Formula, CE = (2a - 0) 2 + (0 - a) 2 = 4a 2 + a 2 = a 5 AD = (a - 0) 2 + (0-2a) 2 = a 2 + 4a 2 = a 5 Thus CE AD by the definition of congruence. Position each figure in the coordinate plane and give the coordinates of each vertex. 22. a right triangle with leg lengths r and s 23. a rectangle with length 2p and width p 24. a square with side length 8m For exercises 25 and 26 assign coordinates to each vertex and write a coordinate proof. 25. Given: In rectangle ABCD, E is the midpoint of AB, F is the midpoint of BC, G is the midpoint of CD, and H is the midpoint of AD. Prove: EF GH 26. Given: PQR has a right Q. M is the midpoint of PR. Prove: MP = MQ = MR 27. Show that a triangle with vertices at (3, 5), (3, 2), and (2, 5) is a right triangle. 4-8 Isosceles and Equilateral Triangles (pp ) EXAMPLE Find the value of x. m D + m E + m F = 180 by the Triangle Sum Theorem. m E = m F by the Isosceles Triangle Theorem. m D + 2 m E = 180 Substitution (3x) = 180 Substitute the given values. 6x = 138 x = 23 Simplify. Divide both sides by 6. Find each value. 28. x 29. RS 30. Given: ACD is isosceles with D as the vertex angle. B is the midpoint of AC. AB = x + 5, BC = 2x - 3, and CD = 2x + 6. Find the perimeter of ACD. Study Guide: Review 287

5 1. Classify ACD by its angle measures. x Ç Classify each triangle by its side lengths. 2. ACD 3. ABC x 4. ABD Î, 5. While surveying the triangular plot of land shown, a surveyor finds that m S = 43. The measure of RTP is twice that of RTS. What is m R? Given: XYZ JKL Identify the congruent corresponding parts. 6. JL? 7. Y? 10. Given: T is the midpoint of PR and SQ. Prove: PTS RTQ {ÎÂ - / 8. L * 9. YZ?? * / - +, 11. The figure represents a walkway with triangular supports. Given that GJ bisects HGK and H K, use AAS to prove HGJ KGJ 12. Given: AB DC, AB AC, DC DB Prove: ABC DCB 13. Given: PQ SR, S Q Prove: PS QR * + -, 14. Position a right triangle with legs 3 m and 4 m long in the coordinate plane. Give the coordinates of each vertex. 15. Assign coordinates to each vertex and write a coordinate proof. Given: Square ABCD Prove: AC BD Find each value. 16. y 17. m S - * xèâ xê Ê Þ Â, Given: Isosceles ABC has coordinates A(2a, 0), B(0, 2b), and C(-2a, 0). D is the midpoint of AC, and E is the midpoint of AB. Prove: AED is isosceles. 288 Chapter 4 Triangle Congruence /

6 FOCUS ON ACT The ACT Mathematics Test is one of four tests in the ACT. You are given 60 minutes to answer 60 multiplechoice questions. The questions cover material typically taught through the end of eleventh grade. You will need to know basic formulas but nothing too difficult. You may want to time yourself as you take this practice test. It should take you about 5 minutes to complete. There is no penalty for guessing on the ACT. If you are unsure of the correct answer, eliminate as many answer choices as possible and make your best guess. Make sure you have entered an answer for every question before time runs out. 1. For the figure below, which of the following must be true? I. m EFG > m DEF II. m EDF = m EFD III. m DEF + m EDF > m EFG (A) I only (B) II only (C) I and II only (D) II and III only (E) I, II, and III 2. In the figure below, ABD CDB, m A = (2x + 14), m C = (3x - 15), and m DBA = 49. What is the measure of BDA? 3. Which of the following best describes a triangle with vertices having coordinates (-1, 0), (0, 3), and (1, -4)? (A) Equilateral (B) Isosceles (C) Right (D) Scalene (E) Equiangular 4. In the figure below, what is the value of y? (F) 49 (G) 87 (H) 93 (J) 131 (K) 136 (F) 29 (G) 49 (H) 59 (J) 72 (K) In RST, RS = 2x + 10, ST = 3x - 2, and RT = 1 x If RST is equiangular, what 2 is the value of x? (A) 2 (B) 5 _ 1 3 (C) 6 (D) 12 (E) 34 College Entrance Exam Practice 289

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8 To receive full credit, make sure all parts of the problem are answered. Be sure to provide a complete explanation for your reasoning. Read each test item and answer the questions that follow. Scoring Rubric: 4 points: The student demonstrates a thorough understanding of the concept, correctly answers the question, and provides a complete explanation. 3 points: The student correctly answers the question but does not show all work or does not provide an explanation. 2 points: The student makes minor errors resulting in an incorrect solution but shows and explains an understanding of the concept. 1 point: The student gives a response showing no work or explanation. 0 points: The student gives no response. Item A What theorem(s) can you use, other than the HL Theorem, to prove that MNP XYZ? Explain your reasoning. Item B Can an equilateral triangle be an obtuse triangle? Explain your answer. Include a sketch to support your reasoning. 5. What should a full-credit response to this test item include? 6. A student wrote this response: Why will this response not receive a score of 4 points? 7. Correct the response so that it receives full credit. Item C An isosceles right triangle has two sides, each with length y + 4. Describe how you would find the length of the hypotenuse. Provide a sketch in your explanation. 1. What should a full-credit response to this test item include? 2. A student wrote this response: 8. A student began trying to find the length of the hypotenuse by writing the following: What score should this response receive? Why? 3. Write a list of the ways to prove triangles congruent. Is the Pythagorean Theorem on your list? 4. Add to the response so that it receives a score of 4-points. Is the student on his way to receiving a 4-point response? Explain. 9. Describe a different method the student could use for this response. Test Tackler 291

9 KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1 4 Multiple Choice Use the diagram for Items 1 and 2. A E B D 1. Which of these congruence statements can be proved from the information given in the figure? AEB CED BAC DAC C ABD BCA DEC DEA 2. What other information is needed to prove that CEB AED by the HL Congruence Theorem? AD AB CB AD BE AE DE CE 3. Which biconditional statement is true? Tomorrow is Monday if and only if today is not Saturday. Next month is January if and only if this month is December. Today is a weekend day if and only if yesterday was Friday. This month had 31 days if and only if last month had 30 days. 4. What must be true if than one point? P, Q, S, and T are collinear. PQ intersects ST at more P, Q, S, and T are noncoplanar. PQ and ST are opposite rays. PQ and ST are perpendicular. 5. ABC DEF, EF = x 2-7, and BC = 4x - 2. Find the values of x. -1 and 5 1 and 5-1 and 6 2 and 3 6. Which conditional statement has the same truth value as its inverse? If n < 0, then n 2 > 0. If a triangle has three congruent sides, then it is an isosceles triangle. If an angle measures less than 90, then it is an acute angle. If n is a negative integer, then n < On a map, an island has coordinates (3, 5), and a reef has coordinates (6, 8). If each map unit represents 1 mile, what is the distance between the island and the reef to the nearest tenth of a mile? 4.2 miles 9.0 miles 6.0 miles 15.8 miles 8. A line has an x-intercept of -8 and a y-intercept of 3. What is the equation of the line? y = -8x + 3 y = _ 8 3 x - 8 y = _ 3 8 x + 3 y = 3x JK passes through points J (1, 3) and K (-3, 11). Which of these lines is perpendicular to JK? y = - 1_ 2 x + _ 1 3 y = -2x - 1 _ 5 y = 1 _ 2 x + 6 y = 2x If PQ = 2 (RS) + 4 and RS = TU + 1, which equation is true by the Substitution Property of Equality? PQ = TU + 5 PQ = TU + 6 PQ = 2 (TU) + 5 PQ = 2 (TU) Which of the following is NOT valid for proving that triangles are congruent? AAA ASA SAS HL 292 Chapter 4 Triangle Congruence

10 Use this diagram for Items 12 and 13. A B C D What is the measure of ACD? What type of triangle is ABC? Isosceles acute Equilateral acute Isosceles obtuse Scalene acute Take some time to learn the directions for filling in a grid. Check and recheck to make sure you are filling in the grid properly. You will only get credit if the ovals below the boxes are filled in correctly. To check your answer, solve the problem using a different method from the one you originally used. If you made a mistake the first time, you are unlikely to make the same mistake when you solve a different way. Gridded Response 14. CDE JKL. m E = (3x + 4), and m L = (6x - 5). What is the value of x? 15. Lucy, Eduardo, Carmen, and Frank live on the same street. Eduardo s house is halfway between Lucy s house and Frank s house. Lucy s house is halfway between Carmen s house and Frank s house. If the distance between Eduardo s house and Lucy s house is 150 ft, what is the distance in feet between Carmen s house and Eduardo s house? 16. JKL XYZ, and JK = 10-2n. XY = 2, and YZ = n 2. Find KL. 17. An angle is its own supplement. What is its measure? 18. The area of a circle is 154 square inches. What is its circumference to the nearest inch? E Short Response 20. Given l m with transversal n, explain why 2 and 3 are complementary. 21. G and H are supplementary angles. m G = (2x + 12), and m H = x. 1 2 a. Write an equation that can be used to determine the value of x. Solve the equation and justify each step. b. Explain why H has a complement but G does not. 22. A manager conjectures that for every 1000 parts a factory produces, 60 are defective. a. If the factory produces 1500 parts in one day, how many of them can be expected to be defective based on the manager s conjecture? Explain how you found your answer. b. Use the data in the table below to show that the manager s conjecture is false. Day Parts Defective Parts n l m BD is the perpendicular bisector of AC. a. What are the conclusions you can make from this statement? b. Suppose BD intersects AC at D. Explain why BD is the shortest path from B to AC. Extended Response 24. ABC and DEF are isosceles triangles. BC EF, and AC DF. m C = 42.5, and m E = 95. a. What is m D? Explain how you determined your answer. b. Show that ABC and DEF are congruent. c. Given that EF = 2x + 7 and AB = 3x + 2, find the value for x. Explain how you determined your answer. 19. The measure of P is 3 1 times the measure of Q. 2 If P and Q are complementary, what is m P in degrees? Cumulative Assessment, Chapters

11 Vocabulary altitude of a triangle centroid of a triangle circumcenter of a triangle circumscribed concurrent equidistant incenter of a triangle indirect proof inscribed locus For a complete list of the postulates and theorems in this chapter, see p. S82. median of a triangle midsegment of a triangle orthocenter of a triangle point of concurrency Pythagorean triple Complete the sentences below with vocabulary words from the list above. 1. A point that is the same distance from two or more objects is? from the objects. 2. A? is a segment that joins the midpoints of two sides of the triangle. 3. The point of concurrency of the angle bisectors of a triangle is the?. 4. A? is a set of points that satisfies a given condition. 5-1 Perpendicular and Angle Bisectors (pp ) Find each measure. JL Because JM MK and ML JK, ML is the perpendicular bisector of JK. JL = KL JL = 7.9 Bisector Thm. Substitute 7.9 for KL. m PQS, given that m PQR = 68 Since SP = SR, SP QP, and SR QR, QS bisects PQR by the Converse of the Angle Bisector Theorem. m PQS = _ 1 m PQR Def. of bisector 2 m PQS = _ 1 (68 ) = 34 Substitute 68 2 for m PQR. Find each measure. 5. BD 6. YZ 7. HT 8. m MNP Write an equation in point-slope form for the perpendicular bisector of the segment with the given endpoints. 9. A (-4, 5), B (6, -5) 10. X (3, 2), Y (5, 10) Tell whether the given information allows you to conclude that P is on the bisector of ABC Chapter 5 Properties and Attributes of Triangles

12 5-2 Bisectors of Triangles (pp ) DG, EG, and FG are the perpendicular bisectors of ABC. Find AG. G is the circumcenter of ABC. By the Circumcenter Theorem, G is equidistant from the vertices of ABC. AG = CG AG = 5.1 Circumcenter Thm. Substitute 5.1 for CG. QS and RS are angle bisectors of PQR. Find the distance from S to PR. S is the incenter of PQR. By the Incenter Theorem, S is equidistant from the sides of PQR. The distance from S to PQ is 17, so the distance from S to PR is also 17. PX, PY, and PZ are the perpendicular bisectors of GHJ. Find each length. 13. GY 14. GP 15. GJ 16. PH UA and VA are angle bisectors of UVW. Find each measure. 17. the distance from A to UV 18. m WVA Find the circumcenter of a triangle with the given vertices. 19. M (0, 6), N (8, 0), O (0, 0) 20. O (0, 0), R (0, -7), S (-12, 0) 5-3 Medians and Altitudes of Triangles (pp ) In JKL, JP = 42. Find JQ. JQ = _ 2 JP Centroid Thm. 3 JQ = 2 _ 3 (42) JQ = 28 Substitute 42 for JP. Multiply. Find the orthocenter of RST with vertices R (-5, 3), S (-2, 5), and T(-2, 0). Since ST is vertical, the equation of the line containing the altitude from R to ST is y = 3. slope of RT = _ (-2) = -1 The slope of the altitude to RT is 1. This line must pass through S (-2, 5). y - y 1 = m (x - x 1 ) Point-slope form y - 5 = 1 (x + 2) Substitution Solve the system y = 3 to find that the y = x + 7 coordinates of the orthocenter are (-4, 3). In DEF, DB = 24.6, and EZ = Find each length. 21. DZ 22. ZB 23. ZC 24. EC Find the orthocenter of a triangle with the given vertices. 25. J (-6, 7), K (-6, 0), L (-11, 0) 26. A (1, 2), B (6, 2), C (1, -8) 27. R (2, 3), S (7, 8), T (8, 3) 28. X (-3, 2), Y (5, 2), Z (3, -4) 29. The coordinates of a triangular piece of a mobile are (0, 4), (3, 8), and (6, 0). The piece will hang from a chain so that it is balanced. At what coordinates should the chain be attached? Study Guide: Review 367