DRILL Name the parts of the figure: 1) All planes parallel to plane ABF Plane DCG 2) All segments that intersect A C B E G H D F AD, CD, GH, AH, EH 3) All segments parallel to AB, GH, EF 4) All segments skew to DH, CG, FG, EH

Parallels § 4.1 Parallel Lines and Planes § 4.2 Parallel Lines and Transversals § 4.3 Transversals and Corresponding Angles § 4.4 Proving Lines Parallel § 4.5 Slope § 4.6 Equations of Lines

Parallel Lines and Transversals What You'll Learn 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 different points is called a __________ transversal B A l m 1 2 4 3 5 6 8 7 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. l m 1 2 3 4 5 7 6 8 Parallel Lines t is a transversal for l and m. t 1 2 3 4 5 7 6 8 b c Nonparallel Lines r is a transversal for b and c. r

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

Parallel Lines and Transversals When a transversal intersects two lines, _____ angles are formed. eight These angles are given special names. l m 1 2 3 4 5 7 6 8 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. Consectutive Interior angles are on the same side of the transversal. ?

Parallel Lines and Transversals Theorem: Alternate Interior Angles If two parallel lines are cut by a transversal, then each pair of Alternate interior angles is _________. congruent 1 2 4 3 5 6 8 7

Parallel Lines and Transversals Theorem: Consecutive Interior Angles If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is _____________. supplementary 1 2 3 4 5 7 6 8

Parallel Lines and Transversals Theorem: Alternate Exterior Angles If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is _________. congruent 1 2 3 4 5 7 6 8 ?

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. Angle 1 and 5 are called __________________. corresponding angles l m 1 2 3 4 5 7 6 8 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: Corresponding Angles If two parallel lines are cut by a transversal, then each pair of corresponding angles is _________. congruent

Transversals and Corresponding Angles Concept Summary Congruent Supplementary Types of angle pairs formed when a transversal cuts two parallel lines. alternate interior consecutive interior alternate exterior corresponding ? 5

Transversals and Corresponding Angles 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 s || t and c || d. Name all the angles that are congruent to 1. Give a reason for each answer. 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 ? 6 – 16

Proving Lines Parallel Postulate 7 – 1 (pg. 364): IF ___________________________________, THEN ________________________________________. two parallel lines are cut by a transversal two parallel lines are cut by a transversal each pair of corresponding angles is congruent each pair of corresponding angles is congruent Converse of that statement (Tomorrow) IF ________________________________________, THEN ____________________________________.

Proving Lines Parallel Postulate 7-1 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 1 2 a b If 1 2, then _____ a || b

Proving Lines Parallel Theorem 7-1 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 are _______. parallel 1 2 a b If 1 2, then _____ a || b

Proving Lines Parallel Theorem 7-2 In a plane, if two lines are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the two lines are _______. parallel 1 2 a b If 1 + 2 = 180, then _____ a || b

Proving Lines Parallel Theorem 7-3 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 are _______. parallel 1 2 a b If 1 2, then _____ a || b

Proving Lines Parallel Theorem 4-8 In a plane, if two lines are perpendicular to a third line, then the two lines are _______. parallel If a  t and b  t, then _____ a b t a || b

Proving Lines Parallel We now have five ways to prove that two lines are parallel. Concept Summary Show that a pair of corresponding angles is congruent. Show that a pair of alternate interior angles is congruent. Show that a pair of alternate exterior angles is congruent. Show that a pair of consecutive interior angles is supplementary. Show that two lines in a plane are perpendicular to a third line.