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Section 5.3 Congruent Angles Associated with Parallel Lines

Section 5.3 Congruent Angles Associated with Parallel Lines. The Parallel Postulate. Through a point not on a line there is exactly one parallel to the given line. Euclidean Geometry treats it as a truth. Hyperbolic Geometry was discovered by trying to prove the parallel postulate.

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Section 5.3 Congruent Angles Associated with Parallel Lines

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  1. Section 5.3 Congruent Angles Associated with Parallel Lines

  2. The Parallel Postulate • Through a point not on a line there is exactly one parallel to the given line.

  3. Euclidean Geometry treats it as a truth. • Hyperbolic Geometry was discovered by trying to prove the parallel postulate. • Spherical Geometry was discovered as another case. • We could not go to the moon without Hyperbolic Geometry.

  4. Theorem 37: If two lines ||, then each pair of alternate interior angles are congruent. 1 3 4 2 <1 and <2, <3 and <4

  5. Theorem 38: If two parallel linesare cut by a transversal then any pair of angles formed are either congruent or supplementary. 1 2 1 3 4 5 6 7 8

  6. Six Theorems about Parallel Lines • Th39: || lines => alt. ext. <s = • Th40: || lines => corr. <s = • Th41: || lines => same side int. <s supp • Th42: || lines =>same side ext.<s supp • Th43: In a plane, if a line is | to one of two || lines, it is | to the other. • Th44: Transitive Prop of || Lines If two lines are || to a third line, they are || to each other.

  7. A Crook Problem • If a||b find the measure of <1. 30 m <1 120 Draw m by || Postulate

  8. A Crook Problem • If a||b find the measure of <1. 30 30 m <1 60 120 Fill in measures of appropriate angles Then <1 is 90º

  9. If a||b find the measures of all the angles. 70 110

  10. If a||b find the measures of all the angles. 110 70 70 110 110 70 70 110

  11. Isosceles Trapezoid • A trapezoid is a four sided figure with one set of parallel sides. • An isosceles trapezoid is a trapezoid with legs congruent.

  12. Sample Problem 5 E C B • Given: Figure ABCD is Isoceles Trapezoid • Prove: <B = <C A D • … • Draw DE||AB • Draw AE • <DAE=<BEA, <BAE=<DEA • AE=AE • Triangles AEB = EAD • AB=DE • DE=DC • <DEC=<C • <B=<DEC • <B = <C • Given • Parallel Postulate • 2 pts determine a line • || lines => alt int <s = • Reflexive • ASA • CPCTC • Transitive • If legs = then base angles =. • || lines => corr. <s = • Transitive

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