Geometry: A Complete Course

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1 Geometry: A Complete Course (with Trigonometry) Module C Instructor's Guide with etailed Solutions for Progress Tests Written by: Larry E. Collins

2 Quiz Form A Class ate Score Unit III - Fundamental Theorems Part A - eduction Proof Lesson 2 - Inirect Proof 1. State the negation of each statement a) It will not rain. It is not true that it will not rain. Or, it will rain. b) ABC is an isosceles triangle. It is not true that ABC is an isoceles triangle. Or, ABC is not an isoceles triangle. c) x + 5 is not an open phrase. It is not true that x + 5 is not an open phrase. Or x + 5 is an open phrase. 2. Indicate whether each pair of statements would enable you to arrive at a contradiction in an indirect proof, and give some justification for your answer. a) AB < 15; AB > 20 Yes. These two statements are already contradictory b) X and Y are obtuse angles; X and Y are supplementary. Yes. X and Y are greater than 90. c) Point B is between points A and C; Points A, B, and C are not collinear. Yes, by the definition of Betweenness d) P and Q are congruent; P and Q are complementary. No. Both can be true without contradiction 3. For each of the following conditionals, state the assumption you would use to start an indirect proof. a) If a triangle is equilateral, then the triangle is isosceles. Suppose (assume) the triangle is not isoceles. b) If an angle is a right angle, then the angle is equal to its supplement. Suppose (assume) the angle is not equal to its supplement VideoTextInteractive Geometry: A Complete Course 9

3 Quiz Form A Class ate Score Unit III - Fundamental Terms Part - Theorems About Segments and Rays Lesson 1 - Theorem 5: If two different lines intersect, then exactly one plane contains both lines. Lesson 2 - Theorem 6: If in a plane, there is a point on a line, then there is exactly one point perpendicular to the line, through that point. 1. In the diagram to the right, BE AC and B BF. Find the measure of each of the following angles. a) mebf 55 O b) mbe 35 O c) mba 55 O d) mbc 125 O 2. In the diagram to the right, BE AC and B BF. Also assume mcbf = x. Express the measure of each of the following angles: a) mebf 90 x b) mbe x c) mba 90 x d) mbc 90 + x = = = = (90 x) or x E F 35 O A B C E F (x) O A B C 3. In the diagram to the right, BE AC and B BF. Find the value of x in each of the following problems. a) mbe = 3x, mebf = 4x 1 x = 13 3x + (4x 1) = 90 7x 1 = 90 7x = 91 x = 13 E (3x) O (4x 1) O F A B C b) mab = 6x, mbe = 3x + 9, x = 9 mebf = 4x + 18, mfbc = 4x 6x + 3x x x = x + 27 = x = 153 x = VideoTextInteractive Geometry: A Complete Course 25

4 Unit III, Part, Lessons 1&2, Quiz Form A 4. Using the format given below, write a complete proof with the Given and Prove information, and the iagram. Given: m1 = m2 and m3 = m4 as shown Prove: YS XZ iagram: R S T X Y Z STATEMENT REASON 1. m1 = m2 1. Given 2. m3 = m4 2. Given 3. m1 + m3 = m2 + m4 3. Addition for Equality 4. m1 + m3 = mxys 4. Postulate 7 (Protractor) - Angle-Addition Assumption 5. m2 + m4 = msyz 5. Postulate 7 (Protractor) - Angle-Addition Assumption 6. mxys = msyz 6. Substitution (from statements 3, 4, and 5) 7. XYZ is a straight angle whose 7. efinition of Straight Angle measure of 180 O 8. mxys + msyz = mxyz 8. Postulate 6 (Ruler) - Angle-Addition Assumption 9. mxys + msyz = Substitution (from statements 7 and 8) 10. mxys + mxys = Substitution (from statements 6 and 9) 11. 2mXYS = Collect like terms (istributivity) 12. XYS = Multiplication for Equality 13. XYS is a right 13. efinition of Right Angle 14. YS XZ 14. efinition of Perpendicular Lines VideoTextInteractive Geometry: A Complete Course

5 Quiz Form A Class ate Score Unit III - Fundamental Terms Part F - Theorems About Segments and Rays Lesson 1 - Theorem 9: If two adjacent acute angles have their exterior sides in perpendicular lines, then the two angles are complementary. Lesson 2 - Theorem 10: If the exterior sides of two adjacent angles are opposite rays, then the angles are supplementary. For each of the following statements 1 through 10, write either true or false. 1. Two angles may be both adjacent and congruent. 2. Two angles may be both complementary and supplementary. False 3. If two angles are acute, they cannot be supplementary. 4. If two lines intersect, then four pairs of supplementary and adjacent angles are formed. 5. If AOB and BOC are supplementary and adjacent, then OA and OC cannot be a pair of opposite rays. False 6. If AOB and BOC are adjacent, then B lies inside AOC. 7. If two angles formed by two lines are adjacent, then they are supplementary. 8. If B lies inside AOC, then AOB and BOC are adjacent. 9. If the measure of an angle is 120, the measure of the complement is 60. False 10. If AOB and BOC are adjacent, then maob + mboc = maoc VideoTextInteractive Geometry: A Complete Course 41

6 Unit III, Part F, Lessons 1&2, Quiz Form A 12. Given: AC BEC as shown Prove: BC AEC A C B E STATEMENT REASON 1. AC and CB are adjacent angles 1. efinition of Adjacent Angle BEC and CEA are adjacent angles 2. A and B are opposite rays 2. efinition of Opposite Rays EB and EA are opposite rays 3. AC and CB are supplementary 3. Theorem 10 - If the exterior sides of two adjacent angles are BEC and CEA are supplementary opposite rays, then the angles are supplementary. 4. mac + mcb = efinition of Supplementary Angles mbec + mcea = mac + mcb = mbec + mcea 5. Substitution of Equals (4 into 4) 6. AC BEC 6. Given 7. mac = mbec 7. efinition of Congruent Angles 8. mac + mcb mac = mbec + 8. Addition Property of Equality mcea mbec 9. mac + mcb + mac = mbec + 9. efinition of Subtraction (a b means a + b) mcea + mbec 10. mac + mac + mcb = mbec Commutative Property of Addition mbec + mcea mcb = 0 + mcea 11. Additive Inverse Property 12. mcb = mcea 12. Identity Property of Addition 13. BC AEC 13. efinition of Congruent Angles 2006 VideoTextInteractive Geometry: A Complete Course 43

7 Quiz Form A Class ate Score Unit III - Fundamental Terms Part F - Theorems About Segments and Rays Lesson 3 - Theorem 11: If you have right angles, then those right angles are congruent. Lesson 4 - Theorem 12: If you have straight angles, then those straight angles are congruent. 1. each of the following using the figure at the right. a) Two pairs of opposite rays. YU and YX; YT and YW T U b) Two right angles. VYW; VYT c) Two straight angles. TYW; XYU X Y W V d) Three acute angles. TYU; UYV; XYW e) Three obtuse angles. XYT; UYW; XYV f) Two points in the exterior of VYT point X; point W g) The sides of XYV YX; YV h) The vertex of all angles. point Y i) A point in the interior of TYV point U j) An angle which is congruent to WYV VYT k) An angle which is congruent to TYW XYU 2006 VideoTextInteractive Geometry: A Complete Course 51

8 Unit III, Part F, Lessons 3&4, Quiz Form A 5. Given: XY is a right angle, ABW is a right angle, XYZ is a straight angle, and ABC is a straight angle as shown Prove: YZ and WBC are supplementary W X Y B A Z C STATEMENT REASON 1. XY is a right angle 1. Given 2. ABW is a right angle 2. Given 3. mxy = efinition of a Right Angle 4. mabw = efinition of a Right Angle 5. XYZ is a straight angle 5. Given 6. ABC is a straight angle 6. Given 7. mxyz = efinition of a Straight Angle 8. mabc = efinition of a Straight Angle 9. mxyz = mxy + myz 9. Postulate 7 (Protractor) Angle-Addition Assumption 10. mabc = mabw + mwbc 10. Postulate 7 (Protractor) Angle-Addition Assumption = 90 + myz 11. Substitution of Equals = 90 + mwbc 12. Substitution of Equals = 90 + myz Subtraction Property for Equality = 90 + mwbc Subtraction Property for Equality = 90 + myz Substitution of Equals = 90 + mwbc Substitution of Equals = 90 + myz efinition of Subtraction = 90 + mwbc efinition of Subtraction = myz 19. Commutative Property of Addition = mwbc 20. Commutative Property of Additio = 0 + myz 21. Additive Inverse Property = 0 + mwbc 22. Additive Inverse Property = myz 23. Identity Property of Addition = mwbc 24. Identity Property of Addition = myz + mwbc 25. Addition Property for Equality = myz + mwbc 26. Substitution of Equals ( = 180) 27. YZ and WBC are supplementary 27. efinition of Supplementary Angles 2006 VideoTextInteractive Geometry: A Complete Course 55

9 Unit III, Part F, Lessons 3&4, Quiz Form B 2. Solve the inequality 2y + 8 < 6y, and graph the solution on a number line. Then answer the following: 2y + 8 < 6y 2y + 2y + 8 < 6y + 2y < 8y 8 < 8y > 1 8y 8 1 > y or y < a) oes the graph of the solution set of the inequality represent a ray? No Explain. the graph does not have a definite endpoint b) Write the inequality whose solution set is the complement of the solution set for 2y + 8 < 6y. 2y + 8 < 6y or 2y + 8 6y c) The union of the two solutions to the inequalities in part a) and part b) is a straight angle VideoTextInteractive Geometry: A Complete Course

10 Quiz Form A Class ate Score Unit III - Fundamental Terms Part G - Theorems About Segments and Rays Lesson 1 - Theorem 13: If two angles are complementary to the same angle or congruent angles, then they are congruent to each other. Lesson 2 - Theorem 14: If two angles are supplementary to the same angle or congruent angles, then they are congruent to each other. Lesson 3 - Theorem 15: If two lines intersect, then the vertical angles formed are congruent. In the diagram, at the right, AFB is a right angle. the figures described in exercises 1 through 6 below. 1. Another right angle AF A 2. Two complementary angles. AFE and EF E 3. Two congruent supplementary angles. BFA and FA CFA and EFA or BFC and FC or 4. Two non-congruent supplementary angles. BFE and EF B C F 5. Two acute vertical angles. BFC and FE 6. Two obtuse vertical angles. CF and BFE In the diagram, at the right, OT bisects SOU, muov = 30 and myow = 126. Find the measure of each angle. 7. mvow 24 O 8. mzoy 30 O T U V 9. mtou 63 O S O W 10. mzow 156 O 11. muos 126 O Z Y X 12. mtoz 87 O 2006 VideoTextInteractive Geometry: A Complete Course 63

11 Unit III, Part H, Lessons 1,2&3, Quiz Form B In the diagram at the right, p q and t r. Use this diagram to find m12 in Exercises 5 through t r p q 5. m2 = 75 O m12 = 75 O 6. m3 = 110 O m12 = 70 O m2 = m12 m3 = m1 m12 + m1 = 180 m = 180 m12 = m16 = 80 O m12 = O m14 = 72 O m12 = 72 O m16 = m12 m14 = m16 m16 = m12 m14 = m12 9. m6 = 68 O m12 = O m5 = 104 O m12 = 76 O m6 = m16 m16 = m12 m6 = m12 m5 + m16 = 180 m16 = m12 m5 + m12 = m12 = 180 m12 = m12 + m16 = 132 O m12 = O m8 + m10 = 162 O m12 = 81 O m12 = m16 m12 = 66 m8 = m16 m16 = m10 m8 = m10 m10 = 81 m10 = m12 m12 = VideoTextInteractive Geometry: A Complete Course

12 Unit III, Part H, Lessons 4,5,6&7, Quiz Form A 13. Given: G AC and ACE is a right angle as shown Prove: ABC and FBG are complementary angle E B F G A C STATEMENT REASON 1. ACE is a right angle 1. Given 2. CE AC 2. efinition of Perpendicular Lines. 3. G AC 3. Given 4. CE G 4. Theorem 18 - If a given line is perpendicular to one of two parallel lines, then it is perpendicular to the other. 5. BC is a right angle 5. efinition of Perpendicular Lines 6. mbc = efinition of Right Angle. 7. mba + mabc = mbc 7. Postulate 7 (Protractor) - Angle-Addition Assumption 8. BA FBG 8. Theorem 15 - If two lines intersect, then the vertical angles formed are congruent 9. mba = mfbg 9. efinition of Congruent Angles 10. mfbg + mabc = mbc 10. Substitution of Equality (9 into 7) 11. mfbc + mabc = Substitution of Equality (6 into 10) 12. ABC and FBC are complementary angles 12. efinition of Complementary Angles 2006 VideoTextInteractive Geometry: A Complete Course 87

13 Unit III, Test Form B (II-C-1,7) 2. Given: collinear points, P, Q, R, and S PQ SR P Q R S Prove: PR SQ STATEMENT REASON 1. Points P, Q, R and S are collinear 1. Given 2. PQ SR 2. Given 3. PQ = SR 3. efinition of Congruent Line Segments 4. QR = RQ 4. Reflexive Property of Equality 5. PQ + QR = SR + RC 5. Addition Property for Equality 6. PQ + QR = PR 6. Postulate 6 (Ruler) - Segment-Addition Assumption 7. SR + RC = SQ 7. Postulate 6 (Ruler) - Segment-Addition Assumption 8. PR = SQ 8. Substitution of Equality (statements 5, 6 and 7) 9. PR SQ 9. efinition of Congruent Line Segments VideoTextInteractive Geometry: A Complete Course

Questions. Exercise (1)

Questions. Exercise (1) Questions Exercise (1) (1) hoose the correct answer: 1) The acute angle supplements. angle. a) acute b) obtuse c) right d) reflex 2) The right angle complements angle whose measure is. a) 0 b) 45 c) 90