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Proving Statements About Segments: Understanding Congruence in Geometry

In this lesson, students will learn to justify statements about congruent segments through two-column proofs and understand essential geometric theorems. Key concepts include the meanings of reflexive, symmetric, and transitive properties of segment congruence, as well as how to apply these principles in problem-solving. Using examples and diagrams, students will practice completing proofs and reinforce their understanding of geometric relationships. This will prepare them for upcoming assessments and deepen their comprehension of geometric concepts.

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Proving Statements About Segments: Understanding Congruence in Geometry

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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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