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2.4 Proving Statements about Segments

This lesson focuses on teaching students geometric concepts by justifying statements about congruent segments and writing logical proofs. Students will learn the essential properties of segment congruence, including reflexive, symmetric, and transitive properties, through two-column proofs and paragraph proofs. They will engage in activities that enhance their understanding of segment relationships, including midpoint concepts and practical segment construction. By the end of the lesson, students will confidently prove segment congruence using various mathematical reasoning techniques.

dylan-robertson
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2.4 Proving Statements about Segments

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  1. 2.4 Proving Statements about Segments

  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. NOTE: Put in the Definitions/Properties/ Postulates/Theorems/Formulas portion of your notebook • 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. Paragraph Proof • A proof can be written in paragraph form. It is as follows: • You are given that PQ ≅ to 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.

  7. 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

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

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

  10. 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.

  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 Definition of a midpoint Segment Addition Postulate Substitution Property Distributive property Division property Substitution Statements: Reasons:

  12. Activity—Copy a segment • Use the following steps to construct a segment that is congruent to AB. • Use a straightedge to draw a segment longer than AB. Label the point C on the new segment. • Set your compass at the length of AB. • Place the compass point at C and mark a second point D on the new segment. CD is congruent to AB.

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