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Parallel lines: the parallel line are the lines that are coplanar meaning that they lie in the same plane and never intersect. In the picture below the orange lines are representing parallel lines. Parallel planes: the parallel planes are the planes that never intersect. In the picture below the light blue and orange represent parallel planes. It is important to mention that their are lines named skew lines these lines are totally the opposite to the parallel lines, the skew lines are not coplanar, not parallel and do not intersect. In the picture The gray and the orange lines Represent skew lines. Parallel postulate: through a point v not on line c there is exactly one line parallel to c. Perpendicular postulate: if 2 lines intersect and they create a linear pair of congruent angles, then the lines are perpendicular. Transversal: the transversals is a line intersecting two parallel lines at two different points. : the corresponding angles lie on the same side of the transversal and in the same sides of the lines in the picture in the next page the transversal would be A and the lines would be T and S. : the alternate exteriors lie on the opposite side of the transversal, these means that they lie outside the two lines. : these are non adjacent angles that lie on the opposite side of the transversal these means that they are inside the two lines and not outside. : these angles lie inside the two lines and lie on the same side of the transversal. 1 Coresponding: 2 and 6 Alternate ex: 2 and 7 Alternate int: 4 and 5 Consecutive: 4 and 6 2 4 3 5 6 7 8 Corresponding angles postulate: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 2. m<def Examples: 1. m<ABC x= 78 M<ABC=78 80 a c b 3. m<def (2x– 6)=(x+3) m<def= 9+3 -x -x 12 X– 6= 3 +6 +6 x=9 (2x-4)=(x+2) -x -x X-4=2 +4 +4 m<def =X=6 6+2=8 2x-6 X+3 2x -- 4 X+2 Postulate of the converse: if two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel. 1. <1 =<5 <1=<5 l m 1 2 4 3 5 7 l 6 8 m 2. m<1=(4x+16), m<8=(5x-12),x=28 1=4(28)+16=128 8=5(28)-12=128 M<1=m<8 M<4=(6x-19), m<5=(3x+14), x=11 L ll m 4=6(11)-19=47 5=3(11)+14=47 M<4=m<5 L ll m Alternate interior angle theorem: if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Converse: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. <2 = <3 (= is congruent) 1 2 1 2 3 4 3 4 <1=<4 1 5 2 6 3 7 8 4 <3=<6 Same side interior angle theorem: if two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are supplementary. converse: if two coplanar lines are cut by a transversal so that pair of same-side interior angles are supplementary, then the two lines are parallel. Alternate exterior angle theorem: If two parallel lines are cut by a transversal, then the two pairs of alternate exterior angles are congruent. converse: if two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel. In a plane, if a transversal is perpendicular to one of two parallel lines then is is perpendicular to the other line.