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 2 3 4 <1 and <2, <3 and <4

5. Theorem 38: If two parallel lines are cut by a transversal then any pair of angles formed are either congruent or supplementary. 1 21 34 56 78

6. Six Theorems about Parallel Lines  Th39: || lines => alt. ext. alt. ext. <s =  Th40: || lines => corr. corr. <s =  Th41: || lines => same side int. same side int. <s supp  Th42: || lines =>same side ext. 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. 30 120 Draw m by || Postulate m <1

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

9.  If a||b find the measures of all the angles. 70 110110

10.  If a||b find the measures of all the angles. 70 110 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  Given: Figure ABCD is Isoceles Trapezoid  Prove: <B = <C A B C D E 1.… 2.Draw DE||AB 3.Draw AE 4.<DAE=<BEA, <BAE=<DEA 5.AE=AE 6.Triangles AEB = EAD 7.AB=DE 8.DE=DC 9.<DEC=<C 10.<B=<DEC 11.<B = <C 1.Given 2.Parallel Postulate 3.2 pts determine a line 4.|| lines => alt int <s = 5.Reflexive 6.ASA 7.CPCTC 8.Transitive 9.If legs = then base angles =. 10.|| lines => corr. <s = 11.Transitive