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2.5 Proving Statements about Segments

2.5 Proving Statements about Segments. Mrs. Spitz Geometry Fall 2004. Standards/Objectives:. Standard 3: Students will learn and apply geometric concepts. Objectives: Justify statements about congruent segments. Write reasons for steps in a proof. Definitions. Theorem:

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2.5 Proving Statements about Segments

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  1. 2.5 Proving Statements about Segments Mrs. Spitz Geometry Fall 2004

  2. Standards/Objectives: Standard 3: Students will learn and apply geometric concepts. Objectives: • Justify statements about congruent segments. • Write reasons for steps in a proof.

  3. Definitions Theorem: A true statement that follows as a result of other true statements. Two-column proof: Most commonly used. Has numbered statements and reasons that show the logical order of an argument.

  4. Theorem and Examples • Theorem 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

  5. Given: PQ ≅ XY Prove XY ≅ PQ Statements: PQ ≅ XY PQ = XY XY = PQ XY ≅ PQ Reasons: Given Definition of congruent segments Symmetric Property of Equality Definition of congruent segments Example 1: Symmetric Property of Segment Congruence

  6. Example 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

  7. ________________ ________________ LK = JK LK ≅ JK JK ≅ JL ________________ Given Given Transitive Property _________________ Given Transitive Property Statements: Reasons:

  8. LK = 5 JK = 5 LK = JK LK ≅ JK JK ≅ JL LK ≅ JL Given Given Transitive Property Def. Congruent seg. Given Transitive Property Statements: Reasons:

  9. Example 3: 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.

  10. Q is the midpoint of PR. PQ = QR PQ + QR = PR PQ + PQ = PR 2 ∙ PQ = PR PQ = ½ PR QR = ½ PR Given Definition of a midpoint Segment Addition Postulate Substitution Property Distributive property Division property Substitution Statements: Reasons:

  11. Plan for next week Thurs/Fri– Continue 2.6 Notes/ HW due following class meeting. Mon/Tues – Review Chapter Thur/Fri – Chapter 2 Exam; Chapter 3 definitions

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