GEOMETRY CHAPTER 2: Deductive Reasoning
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1 GEOMETRY CHAPTER 2: Deductive Reasoning NAME Page 1 of 34
2 Section 2-1: If-Then Statements; Converses Conditional Statement: If hypothesis, then conclusion. hypothesis conclusion converse conditional statement in which the and the are switched. If, then. counterexample an example that refutes or disproves a hypothesis, proposition, or theorem. For #1 to 5, underline the hypothesis once and the conclusion twice of each conditional. 1. VW = XY implies. 2. K is the midpoint of only if JK = KL. 3. n > 8 only if n is greater than I ll dive if you dive. 5. If a = b, then a + c = b + c. If the condition or converse is false, provide a counterexample. 6. If today is Thursday, then tomorrow is Friday. Hypothesis: Conclusion: Converse: Conditional: True or False Converse: True or False Page 2 of 34
3 7. If you have a 95, then you have an A. Hypothesis: Conclusion: Converse: Conditional: True or False Converse: True or False 8. If Lisa lives in Langhorne, then she lives in PA. Hypothesis: Conclusion: Converse: Conditional: True or False Converse: True or False 9. If x = 5, then 4x = 20. Hypothesis: Conclusion: Converse: Conditional: True or False Converse: True or False 5. If x = 2, then x 2 = 4. Hypothesis: Conclusion: Converse: Conditional: True or False Converse: True or False Page 3 of 34
4 Bi Conditional: if and only if or iff if and only if. Ex. Congruent angles are angles that have congruent measures. Ex. Obtuse angles are angles with measures between 90 and 180. Page 4 of 34
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6 Section 2-2: Properties of Algebra Properties of Equality Addition Property Subtraction Property Multiplication Property Division Property Substitution Property Reflexive Property Symmetric Property Transitive Property If a = b and c = d, then. If a = b and c = d, then. If a = b, then. If a = b and c 0, then. If a = b, then either a or b may be for the other in any equation (or inequality). a = If a = b, then. If a = b and b = c, then. Properties of Congruence Reflexive Property D Symmetric Property If, then. If D E, then. Transitive Property If and, then. If D E and E F, then. Properties of Real Numbers Commutative Property Associative Property Distributive Property a + b =, ab = a + (b + c) =, a(bc) = a(b + c) = Page 6 of 34
7 Examples: Justify each step with a Property from Algebra. Follow the example below: Given: 4x 5 = 2 Prove: x = Statements Reasons 1. 4x 5 = 2 1. Given 2. 4x = 3 2. Addition Property of Equality 3. x = 3. Division Property of Equality 1. Given: Prove: a = Statements Reasons Given 2. 3a = a = Given: 11 Prove: z = 40 Statements Reasons Given 2. z + 7 = z = Page 7 of 34
8 3. Given: 15y + 7 = 12 20y Prove: y = Statements Reasons 1. 15y + 7 = 12 20y 1. Given 2. 35y + 7 = y = y = Given: x 2 = Prove: x = 6 Statements Reasons 1. x 2 = 1. Given 2. 5(x 2) = 2x x 10 = 2x x 10 = x = x = 6 6. Page 8 of 34
9 2.2 Substitution Property Practice I. 1. a = b + c 1. Given d = e + f 2. a = d 2. Given Substitution II. 1. a = b + c 1. Given d = e + f 2. b + c = e + f 2. Given Substitution (Diagram is for III. And IV.) D E F III. IV. V. A B C 1. DF = AC 1. Given 2. DE + EF = 2. AB + BC = Substitution 1. DE + EF = AB + BC 1. Given 2. DE + EF = 2. AB + BC = Substitution 1. WOY XOZ 1. Given 2. m WOY = m 1 + m 2 2. m XOZ = m 3 + m Substitution W O X Y Z Page 9 of 34
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11 Section 2-3: Proving Theorems Midpoint Theorem: If M is the midpoint of, then AM = AB and MB = AB. A M B Definition of a Midpoint: If M is the midpoint of, then or AM = MB. Angle Bisector Theorem: If is the bisector of ABC, then m ABX = m ABC and m XBC = m ABC. A X B C Definition of Angle Bisector: If is the bisector of ABC, then ABX XBC or m ABX = m XBC. Reasons Used in Proofs Page 11 of 34
12 Directions Be sure to mark your diagrams as you work. 1. Given: AB = CD Prove: AC = BD A B C D STATEMENTS REASONS 1. AB = CD BC = BC AB + BC = CD + BC AB + BC = AC; CD + BC = BD AC = BD Given: AC = BD Prove: AB = CD A B C D STATEMENTS REASONS Given Segment Addition Postulate Substitution Property Reflexive Property of Equality Subtraction Property of Equality Page 12 of 34
13 3. Given: m 1 = m 2 Prove: m ABD = m CBE A C 1 D B 2 E STATEMENTS REASONS Given 2. m CBD = m CBD Addition Property of Equality 4. m 1 + m CBD = m ABD 4. m 2 + m CBD = m CBE Substitution Property 4. Given: m ABD = m CBE Prove: m 1 = m 2 A C 1 D B 2 E STATEMENTS REASONS 1. m ABD = m CBE Angle Addition Postulate Substitution Property 4. m CBD = m CBD Page 13 of 34
14 Sections 2-1 to Given: m CAT = m BAG Prove: m 1 = m 2 STATEMENTS REASONS 1. m CAT = m BAG m CAT = m 1 + m 3; 2. m BAG = m 2 + m Substitution Property 4. m 3 = m m 1 = m Given: ZP = LC; ZI = OC Prove: IP = LO STATEMENTS REASONS Given Segment Addition Postulate 3. ZI + IP = LO + OC 3. Substitution Property 4. ZI = OC 4. Given Subtraction Property of Equality Page 14 of 34
15 3. Given: m 1 = m 2; m 3 = m 4 Prove: m XYZ = m TUV STATEMENTS REASONS Given 2. m 1 + m 3 = m 2 + m Substitution Property 4. Given: MA = TH Prove: MT = AH STATEMENTS REASONS 1. MA = TH AT = AT Addition Property of Equality Segment Addition Postulate Page 15 of 34
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18 Review Sheet for Quiz 2-1 to If m 1 = 110, then the angle is obtuse. a. hypothesis - b. conclusion - c. Is the statement True or False? If False, give a counterexample. d. converse - e. Is the converse True or False? If False, give a counterexample. 2. If bisects KAT, then m KAL = m LAT. a. hypothesis - b. conclusion - c. Is the statement True or False? If False, give a counterexample. d. converse - e. Is the converse True or False? If False, give a counterexample. Page 18 of 34
19 For questions 3-13, complete the next questions by supplying the letter of the reason. (A) Addition Property of Equality (B) Multiplication Property of Equality (C) Definition of Angle Bisector (D) Angle Addition Postulate (E) Substitution Property (F) Segment Addition Postulate (G) Subtraction Property of Equality (H) Angle Bisector Theorem (I) Reflexive Property (J) Midpoint Theorem (K) Transitive Property (L) Definition of Midpoint Name the definition, postulate, or theorem that justifies each statement. 3. If g = h, then g + k = h + k. 4. MN = MN 5. If x + y = z and x = 6, then 6 + y = z. 6. If PQ = QR and QR = ST, then PQ = ST. 7. If x 5 = 20, then x = If 2 (m 2) = 110, then m 2 = 55. For questions, 9-13, use the diagram to the right. 9. m 3 + m 4 = m MPQ. J 10. If H is the midpoint of, then HM = HJ. 11. If bisects QPM, then m 3 = m QPM. H Q 12. If Q is the midpoint of, then PQ = PJ. 13. PQ + QJ = PJ. M 4 3 P Page 19 of 34
20 COMPLETE THE FOLLOWING PROOF. Given: WX = YZ Prove: WY = XZ W X Y Z STATEMENTS REASONS XY = XY WX + XY = XY + YZ WX + XY = WY 4. XY + YZ = XZ Given: WY = XZ Prove: WX = YZ W X Y Z STATEMENTS REASONS Page 20 of 34
21 Section 2-4: Special Pairs of Angles Complementary Angles: two angles whose measures have a sum of 90. Each angle is called a complement of the other. Supplementary Angles: two angles whose measures have a sum of 180. Each angle is called a supplement of the other Vertical Angles: two angles formed by a pair of intersecting lines that are directly across from one another. **Vertical Angle Theorem: Vertical angles are congruent. Examples: Give the complement and the supplement of each angle measure b Page 21 of 34
22 (3x 5) Classify each statement as true or false. 4. If m A + m B + m C = 180, then A, B, C are supplementary. 5. Vertical angles have the same measure. 6. If 1 and 2 are vertical angles, m 1 = 2x + 18, and m 2 = 3x + 4, then x = 14. Complete with always, sometimes, or never. 7. Vertical angles have a common vertex. 8. Two right angles are complementary. 9. Right angles are vertical angles. 10. Angles A, B, and C are complementary. 11. Vertical angles have a common supplement. 12. If A and B are supplementary, find the value of x, m A, and m B. m A = x + 16, m B = 2x Find the value of x. 70 Page 22 of 34
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26 Section 2-5: Perpendicular Lines Definition of Perpendicular Lines: two lines that intersect to form right angles (90º angles.) If two lines are perpendicular, then they form congruent adjacent angles. If two lines form congruent adjacent angles, then the lines are perpendicular. The exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary. If two angles are supplements of congruent angles (or the same angle), then the two angles are congruent. If two angles are complements of congruent angles (or the same angle, then the two angles are congruent. Page 26 of 34
27 Examples:. Use the diagram to classify each statement as true or false. 1. AB EF 2. CGB is a right angle. 3. CGA is a right angle. 4. m DGB = EGC and EGA are complements. C E A G B F D 6. DGF is complementary to DGA. 7. EGA is complementary DGF. Complete with always, sometimes, or never. 8. Perpendicular lines lie in the same plane. 9. Two lines are perpendicular if and only if they form congruent adjacent angles. 10. Perpendicular lines form 60 angles. 11. If the exterior sides of two adjacent angles are perpendicular, then the angles are supplementary. 12. If a pair of vertical angles are supplementary, the lines forming the angles are perpendicular. Page 27 of 34
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32 Review Sheet for Chapter 2 Test Use the conditional statement to answer questions #1-5. Conditional Statement: If three points are coplanar, then they are collinear. 1. hypothesis: 2. conclusion: 3. Is the conditional statement True or False? (If false, give a counterexample.) 4. Converse: 5. Is the converse True or False? (If false, give a counterexample.) In the diagram,. Name: Y V W X T U 6. two acute vertical angles 7. two congruent supplementary angles 8. two adjacent complementary angles 9. two right angles 10. two obtuse vertical angles Page 32 of 34
33 (A) Addition Property of Equality (L) Subtraction Property of Equality (B) Multiplication Property of Equality (M) Angle Addition Postulate (C) Definition of Angle Bisector (N) Segment Addition Postulate (D) Angle Bisector Theorem (O) Definition of Midpoint (E) Substitution Property (P) Midpoint Theorem (F) Reflexive Property (Q) Transitive Property (G) Definition of complementary angles (R) Definition of perpendicular lines (H) Definition of supplementary angles (S) Vertical angles are congruent (I) If two lines are perpendicular, then they form congruent adjacent angles. (J) If two lines form congruent adjacent angles, then the lines are perpendicular. (K) If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary. For questions #11-20, name the definition, property, postulate, or theorem that justifies each statement. Given: X is the midpoint of ; bisects CXB A D 1 2 X C E B 11. CX + XD = CD 12. m AXC = m DXB 13. AX = XB 14. m 1 = m m 1 = ( )m CXB 16. XE = XE 17. AX = ( ) AB 18. m AXE + m EXB = m BXE + m EXA = m BXA 20. m CXB = m AXD Page 33 of 34
34 In questions #21 25, angles. 21. m ADF = 22. m EDC = 23. m CDB = 24. m ADB = and m 1 = 49. Find the measures of the following G E 1 D F C A B 25. m EDB = In questions #21-23,. Use the given information to find the value of x. 26. m 1 = (7x) ; m 4 = (x + 42) x = 27. m 3 = (4x + 35) ; m 4 = (x + 5) x = 28. m 4 = (7x + 13) ; m EAD = (10x 3) x = 29. m 1 = (x + 1) ; m FAC = (9x 11) x = 30. m 2 = (x + 1) x = 31. COMPLETE THE FOLLOWING PROOF. Given: m CAT = m BAG Prove: m 1 = m 3 A C 1 C E A D F 4 B 2 3 T B G STATEMENTS REASONS Given Angle Addition Postulate 3. m 1 + m 2 = m 2 + m Subtraction Property of Equality Page 34 of 34
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