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Proving Statements About Segments in Geometry

In Mrs. Spitz's Geometry class, students will explore concepts of segment congruence and learn to justify statements using logical reasoning. The lesson covers key principles such as the reflexive, symmetric, and transitive properties of segment congruence. Students will engage in two-column proofs and paragraph proofs, enhancing their understanding of congruent segments through rigorous examples and exercises. Homework includes specific assignments from the text to reinforce these concepts, preparing students for upcoming assessments.

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Proving Statements About Segments 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. Assignment: • pp. 104-106 #1-16 all

  4. 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.

  5. 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

  6. 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

  7. 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.

  8. 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

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

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

  11. 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.

  12. 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:

  13. Pg. 104 – 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.

  14. Plan for next week Monday – 2.6 Notes/ HW due following class meeting. Tues/Wed – Review Chapter 2 – Due at the end of class period Thur/Fri – Chapter 2 Exam; Chapter 3 definitions pg. 128. Copy the following into your definitions/postulates/theorems portion of your binder: pg. 130, 137, 138, 143, 150, 157, 166, 172 – Look for the green boxes.

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