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Chapter 2 Review Reasoning and Proof Chapter 2: Reasoning and Proof 2.1 Use Induc<ve Reasoning 2.2 Analyze Condi<onal Statements 2.3 Apply Deduc<ve Reasoning 2.4 Use Postulates and Diagrams 2.5 Reason Using Proper<es from Algebra 2.6 Prove Statements about Segments and Angles 2.7 Prove Angle Pair Rela<onships Using Induc+ve and Deduc+ve Reasoning When you make a conjecture based on a paMern, you use induc<ve reasoning. You use deduc<ve reasoning to show whether the conjecture is true or false by using facts, deﬁni<ons, postulates, or proven theorems. If you can ﬁnd one counterexample to the conjecture, then you know the conjecture is false. Understanding Geometric Rela+onships in Diagrams The following can be assumed from the diagram: A, B, and C are coplanar. ∠ABH and ∠HBF are a linear pair. !##" Plane T and plane S intersect in BC . !##" CD lies in plane S. ∠ABC and ∠HBF are vertical angles. !##" AB ⊥ plane S. Wri+ng Proofs of Geometric Rela+onship You can write a logical argument to show a geometric rela<onship is true. In a two-­‐column proof, you use deduc<ve reasoning to work from GIVEN informa<on to reach a conjecture you want to PROVE. REVIEW KEY VOCABULARY: • conjecture • induc<ve reasoning • counterexample • condi<onal statement • converse • inverse • contraposi<ve • if-­‐then form • hypothesis • conclusion REVIEW KEY VOCABULARY: • nega<on • equivalent statements • perpendicular lines • bicondi<onal statement • deduc<ve reasoning • proof • two-­‐column proof • theorem VOCABULARY EXERCISES 1. Copy and complete: A statement that can be proven is called a(n) _theorem ________? 2. Compare the inverse of a condi<onal statement to the converse of the condi<onal statement. In the inverse the hypothesis and conclusion are negated while in the converse the hypothesis and conclusion are switched. VOCABULARY EXERCISES 3. You know m∠A = m∠B and m∠B = m∠C. What does the Transitive Property of Equality tell you about the measures of the angles? m∠A = m∠C 2.1 USE INDUCTIVE REASONING Describe the paMern in the numbers 3, 21, 147, 1029, …, and write the next three numbers in the paMern. 2.1 USE INDUCTIVE REASONING 4. Describe the paMern in the numbers -­‐20,480, -­‐5120, -­‐1280, -­‐320, . . . . Write the next three numbers. Each number is ¼ of the previous number; -­‐80, -­‐20, -­‐5 2.1 USE INDUCTIVE REASONING 5. Find a counterexample to disprove the conjecture: If the quo<ent of two numbers is posi<ve, then the two numbers must both be posi<ve. −10 =5 −2 2.2 ANALYZE CONDITIONAL STATEMENTS Write the if-­‐then form, the converse, the inverse, and the contraposi<ve of the statement “Black bears live in North America.” a. If-­‐then form: If a bear is a black bear, then it lives in North America. b. Converse: If a bear lives in North America, then it is a black bear. c. Inverse: If a bear is not a black bear, then it does not live in North America. d. Contraposi<ve: If a bear does not live in North America, then it is not a black bear. 2.2 ANALYZE CONDITIONAL STATEMENTS 6. Write the if-­‐then form, the converse, the inverse, and the contraposi<ve of the statement “An angle whose measure is 34° is an acute angle.” a. If-­‐then form: If an angle measure is 34°, then the angle is an acute angle. b. Converse: If an angle is an acute angle, then its measure is 34°. c. Inverse: If an angle measure is not 34°, then the angle is not an acute angle. d. Contraposi<ve: If an angle is not an acute angle, then its measure is not 34°. 2.2 ANALYZE CONDITIONAL STATEMENTS 7. Is this a valid deﬁni<on? Explain why or why not. “If the sum of the measures of two angles is 90°, then the angles are complementary.” Yes: This is the deﬁni<on for complementary angles 2.2 ANALYZE CONDITIONAL STATEMENTS 8. Write the deﬁni<on of an equiangular polygon as a bicondi<onal statement. The interior angles of a polygon are congruent if and only if the polygon is equiangular 2.3 APPLY DEDUCTIVE REASONING Use the Law of Detachment to make a valid conclusion in the true situa<on. If two angles have the same measure, then they are congruent. You know that m∠A = m∠B. Because m∠A = m∠B satisfies the hypothesis of a true conditional statement, the conclusion is also true. So, ∠A ≅ ∠B. 2.3 APPLY DEDUCTIVE REASONING 9. Use the Law of Detachment to make a valid conclusion in the true situa<on. ! If an angle is a right angle, then the angle measures 90 . ∠B is a right angle. ∠B measures 90 . ! 2.3 APPLY DEDUCTIVE REASONING 10. Use the Law of Syllogism to write the statement that follows from the pair of true statements. If x = 3, then 2x = 6. If 4x = 12, then x = 3. If 4x = 12, then 2x = 6. 2.3 APPLY DEDUCTIVE REASONING 11. What can you say about the sum of any two odd integers? Use induc<ve reasoning to form a conjecture. Then use deduc<ve reasoning to show that the conjecture is true. The sum of two odd integers is even. 7 +1 = 8 2m + 1 and 2n + 1 are odd ( 2m + 1) + ( 2n + 1) = 2m + 2n + 2 = 2(m + n + 1) which is even. 2.4 USE POSTULATES AND DIAGRAMS !!!" ∠ABC, an acute angle, is bisected by BE. Sketch a diagram that represents the given information. 1. Draw ∠ABC, an acute angle, and label points A, B, and C. !!!" 2. Draw angle bisector BE . Mark congruent angles. 2.4 USE POSTULATES AND DIAGRAMS !!!" 12. Straight angle CDE is bisected by DK . Sketch a diagram that represents the given information. 2.4 USE POSTULATES AND DIAGRAMS 13. Which of the following statements cannot be assumed from the diagram? A. A, B, and C are coplanar. !##" B. CD ⊥ plane P C. A, F, and B are collinear. !##" D. Plane M intersects plane P in FH. B 2.5 REASONING USING PROPERTIES FROM ALGEBRA Solve 3x + 2(2x + 9) = -10. Write a reason for each step. 3x + 2(2x + 9) = -10 Write original equa<on 3x + 4x + 18 = -10 Distribu<ve Property 7x + 18 = -10 Simplify 7x = -28 Subtrac<on Property x = -4 Division Property 2.5 REASONING USING PROPERTIES FROM ALGEBRA Solve the equa<on. Write a reason for each step 14.− 9x − 21 = −20x − 87 −9x − 21 = −20x − 87 Write original equa<on 11x − 21 = −87 Addi<on Property 11x = −66 Addi<on Property x = -6 Division Property 2.5 REASONING USING PROPERTIES FROM ALGEBRA Solve the equa<on. Write a reason for each step 15. 15x + 22 = 7x + 62 15x + 22 = 7x + 62 Write original equa<on 8x + 22 = 62 Subtrac<on Property 8x = 40 Subtrac<on Property x=5 Division Property 2.5 REASONING USING PROPERTIES FROM ALGEBRA Solve the equa<on. Write a reason for each step 16. 3(2x + 9) = 30 3(2x + 9) = 30 Write original equa<on 2x + 9 = 10 Division Property 2x = 1 1 x= 2 Subtrac<on Property Division Property 2.5 REASONING USING PROPERTIES FROM ALGEBRA Solve the equa<on. Write a reason for each step 17. 5x + 2(2x − 23) = −154 5x + 2(2x − 23) = −154 Write original equa<on 5x + 4x − 46 = −154 Distribu<ve Property 9x − 46 = −154 Simplify 9x = −108 Addi<on Property x = −12 Division Property 2.6 PROVE STATEMENTS ABOUT SEGMENTS AND ANGLES Prove the Reﬂexive Property of Segment Congruence. GIVEN: AB is a line segment PROVE: AB ≅ AB 1. AB is a line segment 1. Given 2. AB is the length of AB 2. Ruler Postulate 3. AB = AB 3. Reﬂexive Property of Equality 4. AB ≅ AB 4. Deﬁni<on of Congruent Segments 2.6 PROVE STATEMENTS ABOUT SEGMENTS AND ANGLES Name the property illustrated by the statement. 18. If ∠DEF ≅ ∠JKL, then ∠JKL ≅ ∠DEF. Symmetric Property of Congruence 2.6 PROVE STATEMENTS ABOUT SEGMENTS AND ANGLES Name the property illustrated by the statement. 19. ∠C ≅ ∠C Reﬂexive Property of Congruence 2.6 PROVE STATEMENTS ABOUT SEGMENTS AND ANGLES Name the property illustrated by the statement. 20. If MN = PQ and PQ = RS, then MN = RS. Transi<ve Property of Equality 2.6 PROVE STATEMENTS ABOUT SEGMENTS AND ANGLES 21. Prove the Transi<ve Property of Angle Congruence GIVEN: ∠A ≅ ∠B and ∠B ≅ ∠C PROVE: ∠A ≅ ∠C 1. ∠A ≅ ∠B and ∠B ≅ ∠C 1. Given 2. m∠A = m∠B and m∠B = m∠C 2. Deﬁni<on of Congruent Segments 3. m∠A = m∠C 3. Transi<ve Property of Equality 4. ∠A ≅ ∠C 4. Deﬁni<on of Congruent Angles 2.7 PROVE ANGLE PAIR RELATIONSHIPS GIVEN: ∠5 ≅ ∠6 PROVE: ∠4 ≅ ∠7 1. ∠5 ≅ ∠6 1. Given 2. ∠4 ≅ ∠5 2. Ver<cal Angles Congruence Thm 3. ∠4 ≅ ∠6 3. Transi<ve Property of Congruence 4. ∠6 ≅ ∠7 4. Ver<cal Angles Congruence Thm 5. ∠4 ≅ ∠7 5. Transi<ve Property of Congruence 2.7 PROVE ANGLE PAIR RELATIONSHIPS In Exercises 22 and 23, use the diagram at the right. 22. If m∠1 = 114 ! , find m∠2, m∠3, and m∠4. m∠1 = 114 ! m∠3 = 114 ! m∠2 = 180 − 114 = 66 ! ! m∠4 = m∠2 = 66! ! 2.7 PROVE ANGLE PAIR RELATIONSHIPS In Exercises 22 and 23, use the diagram at the right. 23. If m∠4 = 57! , find m∠1, m∠2, and m∠3. m∠4 = 57! m∠2 = 57 ! m∠3 = 180 − 57 = 123 ! ! m∠1 = m∠3 = 123! ! 2.7 PROVE ANGLE PAIR RELATIONSHIPS 24. Write a two-­‐column proof GIVEN: ∠3 and ∠2 are complementary. m∠1+ m∠2 = 90! PROVE: ∠3 ≅ ∠1 1. ∠3 and ∠2 are complementary. m∠1+ m∠2 = 90! 2. m∠3 + m∠2 = 90! 1. Given 2. Deﬁni<on of complementary angles 3. m∠1+ m∠2 = m∠3 + m∠2 3. Transi<ve Property of Equality 4. m∠3 = m∠1 5. ∠3 ≅ ∠1 4. Subtrac<on Property of Equality 5. Transi<ve Property of Congruence HOMEWORK Study for test on Friday Page 130: 1-­‐21