Essential Question How can you prove a mathematical statement?

Transcription

1 .5 TEXS ESSENTIL KNOWLEDGE ND SKILLS Preparing for G.6. G.6. G.6.D G.6.E RESONING To be proficient in math, you need to know and be able to use algebraic properties. Proving Statements about Segments and ngles Essential Question How can you prove a mathematical statement? proof is a logical argument that uses deductive reasoning to show that a statement is true. Writing Reasons in a Proof Work with a partner. Four steps of a proof are shown. Write the reasons for each statement. Given = + Prove =. = +. Given. + = = = 4. Writing Steps in a Proof Work with a partner. Six steps of a proof are shown. omplete the statements that correspond to each reason. Given m = m 3 E D Prove m E = m D 3.. Given. m E = m + m 3. ngle ddition Postulate (Post..4) 3. m E = m + m 3. Substitution Property of Equality 4. m E = 4. ommutative Property of ddition 5. m + m = 5. ngle ddition Postulate (Post..4) Transitive Property of Equality ommunicate Your nswer 3. How can you prove a mathematical statement? 4. Use the given information and the figure to write a proof for the statement. Given is the midpoint of. is the midpoint of D. D Prove = D Section.5 Proving Statements about Segments and ngles 99

2 .5 Lesson What You Will Learn ore Vocabulary proof, p. 00 two-column proof, p. 00 theorem, p. 0 Write two-column proofs. Name and prove properties of congruence. Writing Two-olumn Proofs proof is a logical argument that uses deductive reasoning to show that a statement is true. There are several formats for proofs. two-column proof has numbered statements and corresponding reasons that show an argument in a logical order. In a two-column proof, each statement in the left-hand column is either given information or the result of applying a known property or fact to statements already made. Each reason in the right-hand column is the explanation for the corresponding statement. Writing a Two-olumn Proof Write a two-column proof for the situation in Example 4 from the Section.4 lesson. Given m l = m 3 Prove m D = m E D E 3. m = m 3. Given. m D = m 3 + m. ngle ddition Postulate (Post..4) 3. m D = m + m 3. Substitution Property of Equality 4. m + m = m E 4. ngle ddition Postulate (Post..4) 5. m D = m E 5. Transitive Property of Equality Monitoring Progress Help in English and Spanish at igideasmath.com. Six steps of a two-column proof are shown. opy and complete the proof. Given T is the midpoint of SU. S 7x T 3x + 0 U Prove x = 5. T is the midpoint of SU.. ST TU.. Definition of midpoint 3. ST = TU 3. Definition of congruent segments 4. 7x = 3x Subtraction Property of Equality 6. x = hapter Reasoning and Proofs

3 Using Properties of ongruence The reasons used in a proof can include definitions, properties, postulates, and theorems. theorem is a statement that can be proven. Once you have proven a theorem, you can use the theorem as a reason in other proofs. Theorems Theorem. Properties of Segment ongruence Segment congruence is reflexive, symmetric, and transitive. Reflexive For any segment,. Symmetric If D, then D. Transitive If D and D EF, then EF. Proofs Ex., p. 03; Example 3, p. 0; hapter Review.5 Example, p. 8 Theorem. Properties of ngle ongruence ngle congruence is reflexive, symmetric, and transitive. Reflexive Symmetric For any angle,. If, then. Transitive If and, then. Proofs Ex. 5, p. 8;.5 oncept Summary, p. 0; Ex., p. 03 Naming Properties of ongruence Name the property that the statement illustrates. a. If T V and V R, then T R. b. If JL YZ, then YZ JL. SOLUTION a. Transitive Property of ngle ongruence b. Symmetric Property of Segment ongruence STUDY TIP When writing a proof, organize your reasoning by copying or drawing a diagram for the situation described. Then identify the Given and Prove statements. In this lesson, most of the proofs involve showing that congruence and equality are equivalent. You may find that what you are asked to prove seems to be obviously true. It is important to practice writing these proofs to help you prepare for writing more-complicated proofs in later chapters. Proving a Symmetric Property of ongruence Write a two-column proof for the Symmetric Property of Segment ongruence. Given LM NP L M N P Prove NP LM. LM NP. Given. LM = NP. Definition of congruent segments 3. NP = LM 3. Symmetric Property of Equality 4. NP LM 4. Definition of congruent segments Section.5 Proving Statements about Segments and ngles 0

4 Writing a Two-olumn Proof Prove this property of midpoints: If you know that M is the midpoint of, prove that is two times M and M is one-half. Given M is the midpoint of. Prove = M, M =. M is the midpoint of.. M M. Given M. Definition of midpoint 3. M = M 3. Definition of congruent segments 4. M + M = 4. Segment ddition Postulate (Post..) 5. M + M = 5. Substitution Property of Equality 6. M = 6. Distributive Property 7. M = 7. Division Property of Equality Monitoring Progress Name the property that the statement illustrates.. GH GH Help in English and Spanish at igideasmath.com 3. If K P, then P K. 4. Look back at Example 4. What would be different if you were proving that = M and that M = instead? oncept Summary Writing a Two-olumn Proof In a proof, you make one statement at a time until you reach the conclusion. ecause you make statements based on facts, you are using deductive reasoning. Usually the first statement-and-reason pair you write is given information. Proof of the Symmetric Property of ngle ongruence statements based on facts that you know or on conclusions from deductive reasoning Given Prove.. Given. m = m. Definition of congruent angles 3. m = m 3. Symmetric Property of Equality Definition of congruent angles The number of statements will vary. Remember to give a reason for the last statement. opy or draw diagrams and label given information to help develop proofs. Do not mark or label the information in the Prove statement on the diagram. definitions, postulates, or proven theorems that allow you to state the corresponding statement 0 hapter Reasoning and Proofs

5 .5 Exercises Dynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck. WRITING How is a theorem different from a postulate?. OMPLETE THE SENTENE In a two-column proof, each is on the left and each is on the right. Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, copy and complete the proof. (See Example.) 3. Given PQ = RS Prove PR = QS P Q R S. PQ = RS.. PQ + QR = RS + QR Segment ddition Postulate (Post..) 4. RS + QR = QS 4. Segment ddition Postulate (Post..) 5. PR = QS Given is a complement of. 3 Prove is a complement of is a complement of.. Given m + m = m = m 3 4. Definition of congruent angles Substitution Property of Equality 6. is a complement of In Exercises 5 0, name the property that the statement illustrates. (See Example.) 5. If PQ ST and ST UV, then PQ UV. 6. F F 7. If G H, then H G. 8. DE DE 9. If XY UV, then UV XY. 0. If L M and M N, then L N. PROOF In Exercises and, write a two-column proof for the property. (See Example 3.). Reflexive Property of Segment ongruence (Thm..). Transitive Property of ngle ongruence (Thm..) PROOF In Exercises 3 and 4, write a two-column proof. (See Example 4.) 3. Given GFH GHF Prove EFG and GHF are supplementary. E F 4. Given FG, F bisects and DG. Prove DF G H D G F Section.5 Proving Statements about Segments and ngles 03

6 5. ERROR NLYSIS In the diagram, MN LQ and LQ PN. Describe and correct the error in the reasoning. ecause MN LQ and LQ PN L, then MN PN by the Reflexive Property of Segment Q ongruence (Thm..). P M N 9. WRITING Explain why you do not use inductive reasoning when writing a proof. 0. HOW DO YOU SEE IT? Use the figure to write Given and Prove statements for each conclusion. J N M K L 6. MODELING WITH MTHEMTIS The distance from the restaurant to the shoe store is the same as the distance from the café to the florist. The distance from the shoe store to the movie theater is the same as the distance from the movie theater to the café, and from the florist to the dry cleaners. SHOE STORE restaurant shoe store movie theater Flowers DRY LENERS café florist dry cleaners Use the steps below to prove that the distance from the restaurant to the movie theater is the same as the distance from the café to the dry cleaners. a. State what is given and what is to be proven for the situation. b. Write a two-column proof. 7. RESONING In the sculpture shown, and 3. lassify the triangle and justify your answer. 8. MKING N RGUMENT In the figure, SR and QR. Your friend claims that, because of this, by the Transitive Q Property of Segment ongruence (Thm..). Is your friend correct? Explain your reasoning. S R 3 a. The acute angles of a right triangle are complementary. b. segment connecting the midpoints of two sides of a triangle is half as long as the third side.. RESONING Fold two corners of a piece of paper so their edges match, as shown. a. What do you notice about the angle formed at the top of the page by the folds? b. Write a two-column proof to show that the angle measure is always the same no matter how you make the folds.. THOUGHT PROVOKING The distance from Springfield to Lakewood ity is equal to the distance from Springfield to ettsville. Janisburg is 50 miles farther from Springfield than ettsville. Moon Valley is 50 miles farther from Springfield than Lakewood ity is. Use line segments to draw a diagram that represents this situation. 3. MTHEMTIL ONNETIONS Solve for x using the given information. Justify each step. Given QR PQ, RS PQ P Q x + 5 R S 0 ] 3x Maintaining Mathematical Proficiency Use the figure. (Section.6) 4. is a complement of 4, 5. 3 is a supplement of, and m = 33. Find m 4. and m = 47. Find m Name a pair of vertical angles. Reviewing what you learned in previous grades and lessons hapter Reasoning and Proofs