Chapter 1 Line and Angle Relationships

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Chapter 1 Line and Angle Relationships SECTION 1.1: Sets, Statements, and Reasoning 1. a. Not a statement. b. Statement; true c. Statement; true d. Statement; false 5. Conditional 9. Simple 13. H: The diagonals of a parallelogram are perpendicular. C: The parallelogram is a rhombus. 17. First, write the statement in If, then form. If a figure is a square, then it is a rectangle. 21. True H: A figure is a square. C: It is a rectangle. 25. Induction 29. Intuition 33. Angle 1 looks equal in measure to angle 2. 37. A Prisoner of Society might be nominated for an Academy Award. 41. Angles 1 and 2 are complementary. 45. None 49. Marilyn is a happy person. 53. Not valid 29. Congruent 33. AB 37. x+ x+ 3= 21 2x = 18 x = 9 41. 71 45. 32.7 3 = 10.9 49. N 22 E SECTION 1.3: Early Definitions and Postulates 1. AC 5. 1 m 3.28 ft/m = 1.64 feet 2 9. a. A-C-D b. A, B, C or B, C, D or A, B, D 13. a. m and t b. m and AD or AD and t 17. 2x+ 1+ 3x = 6x 4 + 3= 6x 4 1x = 7 x = 7; AB = 38 21. a. SECTION 1.2: Informal Geometry and Measurement b. 1. AB < CD 5. One; none 9. Yes; no; yes 13. Yes; no 17. a. 3 c. b. 1 2 2 25. A 21. Congruent; congruent 25. No 105

106 Chapter 1: Line and Angle Relationships 29. Given: AB and CD as shown (AB > CD) Construct MN on line l so that MN = AB+ CD 25. x+ y = 180 x = 24 + 2 y x+ y = 180 x 2y = 24 2x+ 2 y = 360 x 2y = 24 3 x = 384 x = 128; y = 52 33. a. No b. Yes c. No d. Yes 37. Nothing 29. x 92 + (92 53) = 90 x 92 + 39 = 90 x 53 = 90 x = 143 33. Given: Obtuse MRP Construct: Rays RS, RT, and RU so that MRP is divided into 4 angles. SECTION 1.4: Angles and Their Relationships 1. a. Acute b. Right c. Obtuse 5. Adjacent 9. a. Yes b. No 13. m FAC + m CAD = 180 FAC and CAD are supplementary. 17. 42 21. x + y = 2x 2y x+ y+ 2x 2y = 64 1x+ 3y = 0 3x 1y = 64 + 2y = 10 + 2y = 78 22 y = 88 y = 4; x = 14 37. It appears that the two sides opposite s A and B are congruent. 41. 90 + x+ x = 360 2x = 270 x = 135 SECTION 1.5: Introduction to Geometric Proof 1. Division Property of Equality or Multiplication Property of Equality 5. Multiplication Property of Equality 9. Angle-Addition Property 13. EG bisects DEF 17. 2x = 10 21. 6x 3= 27 25. 1. 2( x + 3) 7 = 11 2. 2x + 6 7= 11 3. 2x 1= 11 4. 2x = 12 5. x = 6

Section 1.7 107 29. 1. Given 2. If an angle is bisected, then the two angles formed are equal in measure. 3. Angle-Addition Postulate 4. Substitution 5. Distribution Property 6. Multiplication Property of Equality 21. 1. M N P Q on MQ 1. Given 2. MN + NQ = MQ 2. Segment-Addition 3. NP + PQ = NQ Postulate 3. Segment-Addition Postulate 4. MN + NP + PQ = MQ 4. Substitution 25. In space, there are an infinite number of lines which perpendicularly bisect a given line segment at its midpoint. 33. 5 x + 5 y = 5( x+ y) SECTION 1.6: Relationships: Perpendicular Lines 1. 1. Given 2. If 2 s are, then they are equal in measure. 3. Angle-Addition Postulate 4. Addition Property of Equality 5. Substitution SECTION 1.7: The Formal Proof of a Theorem 1. H: A line segment is bisected. C: Each of the equal segments has half the length of the original segment. 5. H: Each is a right angle. C: Two angles are congruent. 9. Given: AB CD Prove: AEC is a right angle. 6. If 2 s are = in measure, then they are. 5. Given: Point N on line s. Construct: Line m through N so that m s. 13. Given: Lines l and m Prove: 1 2 and 3 4 9. Given: Triangle ABC Construct: The perpendicular bisectors of each side, AB, AC, and BC. 17. m 1= m 3 3x+ 10= 4x 30 x = 40; m 1 = 130 21. 1. Given 2. If 2 s are comp., then the sum of their measures is 90. 13. No; Yes; No 17. No; Yes; Yes 3. Substitution 4. Subtraction Property of Equality 5. If 2 s are = in measure, then they are.

108 Chapter 1: Line and Angle Relationships 25. 1. Given 2. ABC is a right. 3. The measure of a rt. =90. 4. Angle-Addition Postulate 6. 1 is comp. to 2. 29. The supplement of an acute angle is obtuse. Given: 1 is supp to 2 2 is an acute Prove: 1 is an obtuse 1. 1 is supp to 2 1. Given 2. m 1+ m 2 = 180 2. If 2 s are supp., the sum of their measures is 180. 3. 2 is an acute 3. Given 4. m 2 = x where 0 < x< 90 4. The measure of an acute is between 0 and 90. 5. m 1+ x = 180 5. Substitution (#4 into #3) 6. x is positive m 1 < 180 6. If a+ p1 = b and p1 is positive, then a< b. 7. m 1 = 180 x 7. Substitution Prop of Eq. (#5) 8. x< 0 < 90 x 8. Subtraction Prop of Ineq. (#4) 9. 90 x< 90 < 180 x 9. Addition Prop. or Ineq. (#8) 10. 90 x < 90 < m 1 10. Substitution (#7 into #9) 11. 90 < m 1 < 180 11. Transitive Prop. of Ineq (#6 & #10) 12. 1 is an obtuse 12. If the measure of an angle is between 90 and 180, then the is obtuse. REVIEW EXERCISES 1. Undefined terms, defined terms, axioms or postulates, theorems. 2. Induction, deduction, intuition 3. 1. Names the term being defined. 2. Places the term into a set or category. 3. Distinguishes the term from other terms in the same category. 4. Reversible 4. Intuition 5. Induction 6. Deduction 7. H: The diagonals of a trapezoid are equal in length. C: The trapezoid is isosceles. 8. H: The parallelogram is a rectangle. C: The diagonals of a parallelogram are congruent. 9. No conclusion 10. Jody Smithers has a college degree. 11. Angle A is a right angle. 12. C 13. RST, S, more than 90. 14. Diagonals are and they bisect each other.

Chapter Review 109 15. 16. 17. 18. 2x+ 15= 3x 2 17 = x x = 17; m ABC = 98 19. 2x+ 5+ 3x 4= 86 + 1= 86 = 85 20. 3x 1= 4x 5 4 = x x = 4; AB = 22 x = 17; m DBC = 47 21. 4x 4+ + 2= 25 9x 2= 25 9x = 27 x = 3; MB= 17 22. 2 CD = BC 2(2x+ 5) = x+ 28 4x+ 10= x+ 28 3x = 18 x = 6; AC = BC = 6 + 28 = 34 23. 7x 21= 3x+ 7 4x = 28 x = 7 m 3 = 49 21 = 28 m FMH = 180 28 152 24. 4x+ 1+ x+ 4 = 180 + 5 = 180 = 175 x = 35 m 4= 35+ 4= 39 25. a. Point M b. JMH c. MJ d. KH 26. 2x 6+ 3(2x 6) = 90 2x 6+ 6x 18= 90 8x 24= 90 8x = 114 x = 14 1 4 1 ( ) m EFH = 3(2x 6) = 3 28 6 2 = 322 1 2 = 67 1 2 27. x+ (40 + 4 x) = 180 + 40 = 180 = 140 x = 28 28. a. 2x+ 3+ 3x 2+ x+ 7= 6x+ 8 b. 6x + 8= 32 6x = 24 x = 4 c. 2x + 3= 2(4) + 3= 11 3x 2= 3(4) 2= 10 x + 7= 4+ 7= 11 29. The measure of angle 3 is less than 50. 30. The four foot board is 48 inches. Subtract 6 inches on each end leaving 36 inches. 4( n 1) = 36 4n 4= 36 4n = 40 n = 10 10 pegs will fit on the board. 31. S 32. S 33. A 34. S 35. N

110 Chapter 1: Line and Angle Relationships 36. 2. 4 P 3. 1 4 4. If 2 s are, then their measures are =. 5. Given 6. m 2= m 3 7. m 1+ m 2= m 4+ m 3 8. Angle-Addition Postulate 9. Substitution 10. TVP MVP 37. Given: KF FH JHK is a right Prove: KFH JHF 1. KF FH 1. Given 2. KFH is a right 2. If 2 segments are, then they form a right. 3. JHF is a right 3. Given 4. KFH JHF 4. Any two right s are. 38. Given: KH FJ G is the midpoint of both KH and FJ Prove: KG GJ 1. KH FJ 1. G is the midpoint of both KH and FJ 2. KG GJ 2. Given If 2 segments are, then their midpoints separate these segments into 4 segments.

Chapter Review 111 39. Given: KF FH Prove: KFH is comp to JHF 1. KF FH 1. Given 2. KFH is comp. to JFH 2. If the exterior sides of 2 adjacent s form rays, then these s are comp. 40. Given: 1 is comp. to M 2 is comp. to M Prove: 1 2 1. 1 is comp. to M 1. Given 2. 2 is comp. to M 2. Given 3. 1 2 3. If 2 s are comp. to the same, then these angles are. 41. Given: MOP MPO OR bisects MOP PR bisects MPO Prove: 1 2 1. MOP MPO 1. Given 2. OR bisects MOP 2. Given PR bisects MPO 3. 1 2 3. If 2 s are, then their bisectors separate these s into four s.

112 Chapter 1: Line and Angle Relationships 42. Given: Prove: 4 6 5 6 1. 4 6 1. Given 2. 4 5 2. If 2 angles are vertical s then they are. 3. 5 6 3. Transitive Prop. 43. Given: Figure as shown Prove: 4 is supp. to 2 1. Figure as shown 1. Given 2. 4 is supp. to 2 2. If the exterior sides of 2 adjacent s form a line, then the s are supp. 44. Given: 3 is supp. to 5 4 is supp. to 6 Prove: 3 6 1. 3 is supp to 5 1. Given 4 is supp to 6 If 2 lines intersect, the vertical angles 2. 2. formed are. 3. 3 6 3. If 2 s are supp to congruent angles, then these angles are. 45. Given: VP Construct: VW such that VW = 4 VP 46. Construct a 135 angle.

Chapter Test 113 47. Given: Triangle PQR Construct: The three angle bisectors. 8. a. right b. supplementary 9. Kianna will develop reasoning skills. 10. 3.2 + 7.2 = 10.4 in. It appears that the three angle bisectors meet at one point inside the triangle. 48. Given: AB, BC, and B as shown Construct: Triangle ABC 11. a. x+ x+ 5= 27 2x + 5= 27 2x = 22 x = 11 b. x + 5= 11+ 5= 16 12. m 4= 35 13. a. x+ 2x 3= 69 3x 3= 69 3x = 72 x = 24 b. m 4 = 2(24) 3 = 45 14. a. m 2 = 137 b. m 2= 43 49. Given: m B = 50 Construct: An angle whose measure is 20. 15. a. 2x 3= 3x 28 x = 25 b. m 1= 3(25) 28= 47 16. a. 2x 3 + 6x 1 = 180 8x 4 = 180 8x = 184 x = 23 50. m 2 = 270 CHAPTER TEST b. m 2 = 6(23) 1 = 137 17. x+ y = 90 18. 1. Induction 2. CBA or B 3. AP+ PB= AB 4. a. point b. line 5. a. Right b. Obtuse 6. a. Supplementary b. Congruent 19. 7. m MNP = m PNQ

114 Chapter 1: Line and Angle Relationships 20. 1. Given 2. Segment-Addition Postulate 3. Segment-Addition Postulate 4. Substitution 21. 1. 2x 3= 17 2. 2x = 20 3. x = 10 22. 1. Given 2. 90 3. Angle-Addition Postulate 4. 90 5. Given 6. Definition of Angle-Bisector 7. Substitution 8. m 1= 45

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