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Chapter 2.7 Proving Segment Relationships Objective: Practice using proofs for geometric relationships by starting with segments Spi.1.4 Check.4.3 Use definitions, basic postulates, and theorems about points, lines, angles, and planes to write/complete proofs and/or to solve problems. Use definitions, basic postulates, and theorems about points, lines, angles, and planes to write/complete proofs Add to your listing Geometric Properties Postulate 2.8 (Ruler Postulate) Postulate 2.9 (Segment Addition Postulate) The points on any line or line segment can be paired with real numbers so that given any two points A and B on a line, A corresponds to zeros and B corresponds to a positive real number. If B is between A and C, then AB + BC = AC or If AB + BC = AC, then B is between A and C AB A Theorem 2.2 BC B AC Congruence of segments is reflexive, symmetric, and transitive. Reflexive Property AB AB Symmetric Property If AB CD, then CD AB Transitive Property If AB CD, and CD EF, then AB EF It's not what you look at that matters, it's what you see. Henry David Thoreau C Use paper to solve Given BC = DE Prove AB + DE = AC 1. 2. 3. Statements BC = DE AB + BC = AC AB + DE = AC 1. 2. 3. Reasons Given Segment Addition Postulate Substitution Use paper to solve Given PR QS Prove PQ RS 1. 2. 3. 4. 5. 6. 7. 8. Statements PR QS PR = QS PQ + QR = PR QR + RS = QS PQ + QR = QR + RS PQ = RS PQ RS 1. 2. 3. 4. 5. 6. 7. Reasons Given Definition of Congruence Segment Addition Postulate Segment Addition Postulate Substitution Subtraction Definition of Congruence Proof with Segment Addition process Prove the following: Given: PQ = RS Prove: PR = QS Statements 1. PQ = RS P 1. Q R Reasons Given 2. PQ + QR = QR + RS 2. Addition Property 3. PQ + QR = PR and 3. Segment Addition Postulate 4. Substitution QR + RS = QS 4. PR = QS S Proof with Segment Addition process Prove the following: Given: PR = QS Prove: PQ = RS Statements 1. PR = QS P 1. Q R Reasons Given 2. PR - QR = QS - QR 2. Subtraction Property 3. PR - QR = PQ and 3. Segment Addition Postulate 4. Substitution QS - QR = RS 4. PQ = RS S Proof with Segment Congruence process J Prove the following: Given: JK KL, HJ GH, KL HJ Prove: GH JK Statements 1. JK KL, KL HJ 1. Reasons Given K L H 2. JK HJ 2. Transitive Property 3. HJ GH 3. Given 4. JK GH 4. Transitive Property 5. GH JK 5. Symmetric Property G Prove the following. Given: AC = AB AB = BX CY = XD Prove: AY = BD Which reason correctly completes the proof? Proof: Statements Reasons 1. AC = AB, AB = BX 1. Given 2. AC = BX 2. Transitive Property 3. CY = XD 3. Given 4. AC + CY = BX + XD 4. Addition Property 5. AC + CY = AY; BX + XD = BD ? Addition Postulate Segment 5. ________________ 6. AY = BD 6. Substitution Practice Assignment Page 145, 4-16 even