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2.5 Proving Statements about Segments

2.5 Proving Statements about Segments. Objectives:. To justify statements about congruent segments. Write reasons for steps in a proof. Definitions. Theorem - A true statement that is proven as a result of other true statements. Two-column proof -

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2.5 Proving Statements about Segments

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  1. 2.5 Proving Statements about Segments

  2. Objectives: • To justify statements about congruent segments. • Write reasons for steps in a proof.

  3. Definitions Theorem - A true statement that is proven as a result of other true statements. Two-column proof - Type of proof that has numbered statements and reasons that show the logical order of an argument. Also called a direct proof.

  4. Components of 2-column proofs Some sort of diagram will be here. • Given: “hypothesis” • Prove: “conclusion” Statements Reasons 1. Given information (usually) 1. Given (usually) 2. next logical step 2. geometry “reason” to justify 3. cont. deductive reasoning 3. cont. geometry “reasons” 4. until you prove “conclusion” 4. final geometry “reason” Defs, props, posts, or thms

  5. Properties of Segment Congruence • Thm 2.1 • Segment congruence is reflexive, symmetric, and transitive. • Examples: • Reflexive: For any segment AB, AB ≅ AB • Symmetric: If AB ≅ CD, then CD ≅ AB • Transitive: If AB ≅ CD, and CD ≅ EF, then AB ≅ EF

  6. Given: PQ ≅ XY Prove: XY ≅ PQ Statements PQ ≅ XY PQ = XY XY = PQ XY ≅ PQ Reasons: Given Def. of  segments Symmetric Property Def. of  segments Ex1: Write a 2-column proof for the symmetric property of congruence. Diagram not needed this time.

  7. Another type of proof… • Paragraph proof: A proof written in paragraph form. • Here is an example of the symmetric property of congruence in paragraph form. • You are given that PQ ≅ XY. By the definition of congruent segments, PQ = XY. By the symmetric property of equality, XY = PQ. Therefore, by the definition of congruent segments, it follows that XY ≅ PQ.

  8. Ex 2: Using Congruence • Use the diagram and the given information to complete the missing steps and reasons in the proof. • GIVEN: LK = 5, JK = 5, JK ≅ JL • PROVE: LK ≅ JL

  9. ________________ LK = JK LK ≅ JK JK ≅ JL ________________ 1. Given 2. _________________ 3. _________________ 4. Given 5. Transitive Property Statements: Reasons

  10. Ex3: Using Segment Relationships • In the diagram, Q is the midpoint of PR. Show that PQ and QR are equal to ½ PR. • Given: Q is the midpoint of PR. • Prove: PQ = ½ PR and QR = ½ PR. I don’t really have room for this proof here so I am going to finish it on the next slide. However, proofs are always “package deals” with the given, prove, and the diagram. You must write it all in your assignments.

  11. Q is the midpoint of PR. PQ = QR PQ + QR = PR PQ + PQ = PR 2 ∙ PQ = PR PQ = ½ PR QR = ½ PR Given Def. of midpoint Segment Addition Substitution Simplify Division Substitution Statements Reasons

  12. Assignment • HW 2.5 p. 104 #1-11, 16-19, 28-40 even

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