2-1 Using Inductive Reasoning to Make Conjectures
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1 CHAPTER 2 Chapter Review 2-1 Using Inductive Reasoning to Make Conjectures Find the next term in each pattern. 1. 6, 12, 18, January, April, July, The table shows the score on a reaction time test given to five students in both the morning and afternoon. The lower scores indicate a faster reaction time. Use the table to make a conjecture about reaction times. Student Morning Afternoon Ann Betsy Carla Denise Ellen Show that the conjecture If a number is a multiple of 5, then it is an odd number is false by finding a counterexample. 2-2 Conditional Statements 5. Identify the hypothesis and conclusion of the conditional statement Two angles whose sum is 90 are complementary angles. Write a conditional statement from each of the following. 6. Integers Even Numbers 7. An angle that measures 90 is a right angle. 43 Geometry
2 Determine if each conditional is true. If false, give a counterexample. 8. If an angle has a measure of 90, then it is an acute angle. 9. If 6x 2 4x 12, then x Write the converse, inverse, and contrapositive of the statement If a number is divisible by 4, then it is an even number. Find the truth value of each. converse: truth value: inverse: truth value: contrapositive: truth value: 2-3 Using Deductive Reasoning to Verify Conjectures 11. Determine if the following conjecture is valid by the Law of Detachment. Given: Nicholas can watch 30 minutes of television if he cleans his room first. Nicholas cleans his room. Conjecture: Nicholas watches 30 minutes of television. 12. Determine if the following conjecture is valid by the Law of Syllogism. Given: If a point A is on MN, then it divides MN into MA and AN. If MA AN then A is the midpoint of MN. Conjecture: If a point is on MN, then A is the midpoint of MN. 2-4 Biconditional Statements and Definitions 13. For the conditional If two angles are complementary, then the sum of the measures is 90, write the converse and a biconditional statement. Converse: Biconditional statement: 44 Geometry
3 14. Determine if the biconditional A point divides a segment into two congruent segments if and only if the point is the midpoint of the segment, is true. If false, give a counterexample. 2-5 Algebraic Proof Solve each equation. Write a justification for each step. x 15. m m Identify the property that justifies each statement. 18. m 1 m 2, so m MN PQ, so PQ MN m 3 m 2 m AB CD and CD EF, 21. m A m A so AB EF 2-6 Geometric Proof 22. Fill in the blanks to complete the two-column proof. m MOP m ROP 90 Given: 1 4 Prove: 2 3 Proof: M O R N P Q 45 Geometry
4 1. Statements 1. Given Reasons 2. m 1 m m 1 m 2 m MOP m 3 m 4 m MOP 5. m 1 m 2 m 3 m Transitive Property of Equality Subtraction Property of Equality 23. Use the given plan to write a two-column proof. Given: MOP NOQ Prove: MON POQ Plan: By the definition of angle O congruence, m MOP m NOQ. Use the angle addition postulate to show that m MOP m MON m NOP. Show a similar statement for NOQ. Use the given fact to equate m MON m NOP and m POQ m NOP. The subtraction property of equality allows you to show m MON m POQ. Use the definition of congruent triangles to establish what needs to be proved. M N P Q Statements Reasons 46 Geometry
5 2-7 Flowchart and Paragraph Proofs Use the given two-column proof to write the following. Given: 1 4 Prove: 1 is supplementary to k m 1 m 4 Statements 3. 1 & 2 are supplementary. 3 & 4 are supplementary. 4. m 1 m m 3 m m 1 m 2 m 3 m 4 6. m 1 m 2 m 3 m 1 7. m 2 m 3 8. m 1 m is supplementary to Given Reasons 2. Definition of Congruent Angles 3. Linear Pair Theorem 4. Definition of Supplementary Angles 5. Transitive Property of Equality 6. Substitution Property of Equality 7. Subtraction Property of Equality 8. Substitution Property of Equality 9. Definition of Supplementary s 24. a flowchart proof 25. a paragraph proof 47 Geometry
6 CHAPTER 2 Postulates and Theorems Theorem Theorem Theorem Theorem Theorem Theorem Theorem (Linear Pair Theorem) If two angles form a linear pair, then they are supplementary. (Congruent Supplements Theorem) If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent. (Right Angle Congruence Theorem) All right angles are congruent. (Congruent Complements Theorem) If two angles are complementary to the same angle (or to two congruent angles), then the two angles are congruent. (Common Segments Theorem) If AB CD, then AC BD. (Vertical Angles Theorem) Vertical angles are congruent. If two congruent angles are supplementary, then each angle is a right angle. 48 Geometry
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