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Parallel Lines ;) Two lines that do not intersect are either parallel or skew. Parallel Lines (II lines) are coplanar lines that do not intersect. Skew Lines are non coplanar lines. Therefore, they are neither parallel nor intersecting. Parallel Segments/Rays Segments and rays contained in parallel lines are also called parallel. If two segments do not have an intersection point, but are contained in intersecting lines, we do not call the segments parallel. Parallel Planes Two planes that do not intersect are called parallel planes. A line and a plane are parallel if they do not intersect. Exercises! Exercises! Exercises! Exercises! Exercises! Theorem: If two parallel planes are cut by a third plane, then the lines of intersection are parallel. Given: Plane A ‖ Plane B Plane A intersects plane C in line x Plane B intersects plane C in line y Prove: line x ‖ line y Statement Reason Transversal A line that intersects two or more coplanar lines at different points 1 4 5 8 6 7 2 3 Angles Formed by Two Lines and a Transversal 1 4 5 8 6 2 1) Corresponding angles (corr <s) – two angles that have corresponding positions relative to the two lines. 3 2) Alternate interior angles (alt. int. <s) – two nonadjacent interior angles on opposite sides of the transversal. 7 3) Same-side Interior (SSI <s) – two interior angles on the same side of the transversal. Learning Log Summary LT 1 - I can classify pairs of angles and lines using appropriate vocabulary. Parallel and skew lines are different because… Pairs of angles formed when two lines are intersected by a transversal are… Closure Homework pg. 76 ~ 1-39 (odd) Theorem 3-9 Through a point outside a line, there is exactly one line perpendicular to the given line. Group Task Prove one of the relationships that exist among angles formed when parallel lines are cut by a transversal. • What is a transversal? • What is the difference between inductive and deductive reasoning? • Use inductive reasoning to determine what relationship exists between corresponding angles. Corresponding Angle Postulate If l is parallel to m, then 1 2 . Why is this a postulate instead of a theorem? Group Task Gallery Walk Alternate Interior Angle Theorem If l is parallel to m, then… Same Side Interior Angle Theorem If l is parallel to m, then… Theorem 3-4 (Perpendicular Transversal Thm) If k is parallel to l and k t, then… Learning Log Summary LT 2 - I can use previous definitions, properties and postulates to prove angle relationships when parallel lines are cut by a transversal. The postulate necessary to prove other angle relationships is… To prove relationships among corr/alt int angles… If a transversal is perpendicular to one parallel line, then… Theorem 3-8 Through a point outside a line, there is exactly one line parallel to the given line. Corresponding Angle Postulate If l is parallel to m, then corresponding angles are congruent. Why is this a postulate instead of a theorem? Converse to Corresponding Angle Postulate If corresponding angles are congruent, then l is parallel to m. Why is this a postulate instead of a theorem? Converse to Alternate Interior Angle Theorem Record it on your list! Converse to Same-Side Interior Angle Theorem Record it on your list! Converse to Perpendicular Transversal Theorem Record it on your list! Converse to Perpendicular Transversal Theorem Learning Log Summary LT 4 - I can use previous definitions, properties and postulates to prove that lines are parallel, given specific relationships that exist between the angles formed. The ways to prove that two lines are parallel… The angle relationship you establish also establishes the parallel lines and the transversal because…