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Theorem 35

Theorem 35 . If two lines are cut by a transversal such that two exterior angels on the same side of the transversal are supplementary, the lines are parallel. Proof of Theorem 35. Given: <1 is supp to <7. Statements. Reasons. <1 is supp to <7 <6 is congruent to <7 <EBF is a straight angle

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Theorem 35

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  1. Theorem 35 If two lines are cut by a transversal such that two exterior angels on the same side of the transversal are supplementary, the lines are parallel.

  2. Proof of Theorem 35 Given: <1 is supp to <7 Statements Reasons • <1 is supp to <7 • <6 is congruent to <7 • <EBF is a straight angle • <1 is supp to <2 • <2 is congruent to <7 • <2 is congruent to <3 • BC is parallel to AD • Given • Vertical <s are congruent • Assumed • Supp <s form st. <s • Supp of the same < are congruent • Transitive Property • Alternate interior <s that are congruent form parallel lines

  3. Is M parallel to P? No, because theorem 35 states that exterior angels on the same side must be supplementary not congruent, a contradiction of the picture.

  4. Proof from the Black Book Given: BC is perpendicular to DC, <ADC = 90° Prove: AD is parallel to BC Statements Reasons 1) BC is perpendicular to DC <ADC = 90° 2) <BCF is a rt. <, <BCD is a rt. < 3) <ADC is a rt. < 4) <ADE is a rt. < 5)<EDA supp. <ADC 6) <BCF is congruent to <ADC 7) <ADE supp. <BCF 8) AD is parallel to BC • Given • Perpendicular lines form rt. <s • Rt. <s = 90° • 2 rt. <s form a st. < • Supp. <s form st. <s • Rt. <s are congruent • Substitution • If there are exterior supplementary <s on the same side, then the lines are parallel

  5. Proof from the text book Given: <1 is supplementary to <7 Prove: A is parallel to B Reasons Statements • <1 supp <7 • A is parallel to B • Given • If exterior <s on the same side are supp, then the lines are parallel

  6. Proof form the text book Given: AB is perpendicular to BD CD is perpendicular to BD Prove: AB is parallel to CD Statements Reasons • Given • Perpendicular lines from rt. <s • rt. <s are congruent • Assumed • Supplements form st. <s • Supplements of congruent <s are congruent • If two exterior <s on the same side are supp, then the lines are parallel • AB is perpendicular to BD and CD is perpendicular to BD • <1 and <2 are rt. <s • <1 is congruent to <2 • <BDE is a st. < • <2 supp <CDE • <3 supp <1 • AB is parallel to CD

  7. Euclid“The Father of Geometry”

  8. Euclid 323 – 285 B.C. • Euclid’s book, The Elements, is divided into thirteen books. Parallel lines are mentioned in Book 1. • He studied in Plato’s Academy in Athens, Greece and taught in Alexandria, Egypt. • His work is based on that of Eudoxus and Theaetetus. • He was not the only one to work with geometry. The reason his work is used today is because of the way it is organized. He started with the most basic concepts and worked up to harder more complex ideas.

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