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Parallels § 4.2 Parallel Lines and Transversals § 4.3 Transversals and Corresponding Angles § 4.4 Proving Lines Parallel Parallel Lines and Planes You will learn to describe relationships among lines, parts of lines, and planes. In geometry, two lines in a plane that are always the same parallel lines distance apart are ____________. No two parallel lines intersect, no matter how far you extend them. Parallel Lines and Transversals You will learn to identify the relationships among pairs of interior and exterior angles formed by two parallel lines and a transversal. Parallel Lines and Transversals In geometry, a line, line segment, or ray that intersects two or more lines at transversal different points is called a __________ A 2 1 4 5 8 6 7 3 l m B AB is an example of a transversal. It intercepts lines l and m. Note all of the different angles formed at the points of intersection. Parallel Lines and Transversals Definition of Transversal In a plane, a line is a transversal iff it intersects two or more Lines, each at a different point. The lines cut by a transversal may or may not be parallel. Parallel Lines Nonparallel Lines l 1 2 4 3 lm t 1 2 4 3 m 5 6 8 7 c 5 6 8 7 b || c t is a transversal for l and m. b r r is a transversal for b and c. Parallel Lines and Transversals Two lines divide the plane into three regions. The region between the lines is referred to as the interior. The two regions not between the lines is referred to as the exterior. Exterior Interior Exterior Parallel Lines and Transversals eight angles are formed. When a transversal intersects two lines, _____ These angles are given special names. l 1 2 4 3 m 5 6 8 7 t Interior angles lie between the two lines. Exterior angles lie outside the two lines. Alternate Interior angles are on the opposite sides of the transversal. Alternate Exterior angles are on the opposite sides of the transversal. Same Side Interior angles are on the same side of the transversal. Parallel Lines and Transversals Theorem 3-1 If two parallel lines are cut by a transversal, then each pair of congruent Alternate Alternate interior angles is _________. Interior Angles 1 2 4 3 5 6 8 7 4 6 3 5 Parallel Lines and Transversals Theorem 3-2 If two parallel lines are cut by a transversal, then each pair of supplementary Same Side Same side interior angles is _____________. Interior Angles 1 2 4 3 5 6 8 7 4 5 180 3 6 180 Parallel Lines and Transversals Theorem 3-3 If two parallel lines are cut by a transversal, then each pair of congruent Alternate alternate exterior angles is _________. Exterior Angles 1 2 4 3 5 6 8 7 1 7 2 8 Transversals and Corresponding Angles You will learn to identify the relationships among pairs of corresponding angles formed by two parallel lines and a transversal. Transversals and Corresponding Angles When a transversal crosses two lines, the intersection creates a number of angles that are related to each other. Note 1 and 5 below. Although one is an exterior angle and the other is an interior angle, both lie on the same side of the transversal. corresponding angles Angle 1 and 5 are called __________________. l 1 2 4 3 m 5 6 8 7 t Give three other pairs of corresponding angles that are formed: 4 and 8 3 and 7 2 and 6 Transversals and Corresponding Angles Postulate 3-1 If two parallel lines are cut by a transversal, then each pair of congruent Corresponding corresponding angles is _________. Angles Transversals and Corresponding Angles Types of angle pairs formed when a transversal cuts two parallel lines. Concept Summary Congruent alternate interior alternate exterior corresponding Supplementary Same side interior Transversals and Corresponding Angles s s || t and c || d. Name all the angles that are congruent to 1. Give a reason for each answer. 1 2 5 6 9 10 13 14 3 1 corresponding angles 6 1 vertical angles 8 1 alternate exterior angles 9 1 corresponding angles 14 1 alternate exterior angles 11 9 1 corresponding angles 16 14 1 corresponding angles t 3 7 11 12 15 16 c 4 8 d Proving Lines Parallel You will learn to identify conditions that produce parallel lines. Reminder: In Chapter 1, we discussed “if-then” statements (pg. 24). hypothesis and the Within those statements, we identified the “__________” conclusion “_________”. I said then that in mathematics, we only use the term “if and only if” if the converse of the statement is true. Proving Lines Parallel Postulate 3 – 1 (pg. 116): two parallel lines are cut by a transversal IF ___________________________________, each pair of corresponding angles is congruent THEN ________________________________________. The postulates used are the converse of postulates that you already know. COOL, HUH? , Postulate 3 – 2 (pg. 122): IF ________________________________________, THEN ____________________________________. Proving Lines Parallel In a plane, if two lines are cut by a transversal so that a pair of corresponding angles is congruent, then the lines are parallel _______. Postulate 3-2 1 2 If 1 2, a b a || b then _____ Proving Lines Parallel In a plane, if two lines are cut by a transversal so that a pair of alternate interior angles is congruent, then the two lines parallel are _______. Theorem 3-3 If 1 2, a 2 1 b a || b then _____ Proving Lines Parallel In a plane, if two lines are cut by a transversal so that a pair of alternate exterior angles is congruent, then the two lines parallel are _______. Theorem 3-6 1 2 If 1 2, a b a || b then _____ Proving Lines Parallel In a plane, if two lines are cut by a transversal so that a pair of same side interior angles is supplementary, then the two parallel lines are _______. Theorem 3-4 If 1 + 2 = 180, 1 2 a b a || b then _____ Proving Lines Parallel In a plane, if two lines are cut by a transversal so that a pair of same side interior angles is supplementary, then the two parallel lines are _______. Theorem 3-5 If a t and b t, t a b a || b then _____ Proving Lines Parallel We now have five ways to prove that two lines are parallel. Show that a pair of corresponding angles is congruent. Show that a pair of alternate interior angles is congruent. Concept Summary Show that a pair of alternate exterior angles is congruent. Show that a pair of same side interior angles is supplementary. Show that two lines in a plane are perpendicular to a third line. Proving Lines Parallel Identify any parallel segments. Explain your reasoning. GY and RD are both perpendicu lar to GA therefore, GY RD by Theorem 4 - 8. G R A Y 90° 90° D Proving Lines Parallel B Find the value for x so BE || TS. T (6x - 26)° (2x + 10)° (5x + 2)° ES is a transversal for BE= and mBES + mEST 180TS. (2x + and 10) EST + (5x are + 2)_________________ = 180 side interior angles. (same BES 7x + 12 = 180 If mBES + mEST = 180, then BE || TS by Theorem 4 – 7. 7x = 168 x = 24 Thus, if x = 24, then BE || TS. S E