Proving Segment Relationships Chapter 2.7 Proving Segment Relationships Objective: Practice using proofs for geometric relationships by starting with segments Spi.1.4 Use definitions, basic postulates, and theorems about points, lines, angles, and planes to write/complete proofs and/or to solve problems. Check.4.3 Use definitions, basic postulates, and theorems about points, lines, angles, and planes to write/complete proofs

If AB  CD, and CD  EF, then AB  EF Geometric Properties Add to your listing Postulate 2.8 (Ruler 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. Postulate 2.9 (Segment Addition Postulate) If B is between A and C, then AB + BC = AC or If AB + BC = AC, then B is between A and C Theorem 2.2 Congruence of segments is reflexive, symmetric, and transitive. AB BC A B C AC 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

Use paper to solve Given BC = DE Prove AB + DE = AC Statements Reasons AB + BC = AC AB + DE = AC Given Segment Addition Postulate Substitution

Use paper to solve Given PR  QS Prove PQ  RS Statements Reasons PQ + QR = PR QR + RS = QS PQ + QR = QR + RS PQ = RS PQ  RS Given Definition of Congruence Segment Addition Postulate Substitution Subtraction

Proof with Segment Addition process Prove the following: Given: PQ = RS Prove: PR = QS P Q R S Statements Reasons PQ = RS PQ + QR = QR + RS PQ + QR = PR and QR + RS = QS PR = QS Given Addition Property Segment Addition Postulate Substitution

Proof with Segment Addition process Prove the following: Given: PR = QS Prove: PQ = RS P Q R S Statements Reasons PR = QS PR - QR = QS - QR PR - QR = PQ and QS - QR = RS PQ = RS Given Subtraction Property Segment Addition Postulate Substitution

Proof with Segment Congruence process J Prove the following: Given: JK  KL, HJ  GH, KL  HJ Prove: GH  JK K L Statements Reasons H G JK  KL, KL  HJ JK  HJ HJ  GH JK  GH GH  JK Given Transitive Property Symmetric Property

Given: AC = AB AB = BX CY = XD Prove: AY = BD Prove the following. Given: AC = AB AB = BX CY = XD Prove: AY = BD 1. Given AC = AB, AB = BX 1. 2. Transitive Property AC = BX 2. 3. Given CY = XD 3. 4. Addition Property AC + CY = BX + XD 4. AY = BD 6. Substitution 6. Proof: Statements Reasons Which reason correctly completes the proof? 5. ________________ AC + CY = AY; BX + XD = BD 5. ? Segment Addition Postulate

Practice Assignment Page 145, 4-16 even