2.5 Proving Statements about Segments

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.