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Download Math 53 Winter Q09 2.1 The Parallel Postulate and Special Angles
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Transcript
Math 53 "Winter ’09" 2.1 "The Parallel Postulate and Special Angles" ————————————————————————————————————————————————— Objectives: * Develop more properties of parallel and perpendicular lines and the angles associated with them. ————————————————————————————————————————————————— Key Concepts: Perpendicular Planes Parallel Lines/Planes Parallel Postulate Transversal Interior/Exterior/Corresponding/Alternate Interior/Alternate Exterior Angles ————————————————————————————————————————————————— Preliminaries: De…nition: "Perpendicular Lines" kPerpendicular lines are two lines that meet to form congruent adjacent anglesk Construction 6: [Constructing the line that is perpendicular to a given line from a point not on the given line] Given: A line l and a point P not on l ! Construct: PQ ? l The term perpendicular includes line-ray, line-plane, and plane-plane relationship. Page: 1 Bibiana Lopez Elementary Geometry by Alexander and Koeberlein 2.1 We obtain the following Theorem: Theorem 2.1.1: kFrom a point not on a given line, there is exactly one line perpendicular to the given linek Parallel Lines De…nition: "Parallel Lines" kParallel lines are lines in the same plane that do not intersectk Question: If two parallel planes M and N are intersected by a third plane G. How must the lines of intersection a and b be related? Euclidean Geometry The type of geometry found in this Elementary Geometry textbook is known as Euclidean geometry. In this geometry, a plane is ‡at, two-dimensional surface in which the line segment joining any two points of the plane lies entirely within the plane. Postulate 10: (Parallel Postulate) Through a point not on a line, exactly one line is parallel to the given line Page: 2 Bibiana Lopez Elementary Geometry by Alexander and Koeberlein 2.1 A transversal is a line that intersects two (or more) other lines at distinct points; all of the lines lie in the same plane. t 1 2 3 4 r s 5 7 6 8 Using the picture above, we obtain the following Angles between r and s are interior angles Angles outside r and s are exterior angles Two angles that lie in the same relative positions (such as above and left) are called corresponding angles Two interior angles that have di¤erent vertices and lie on opposite sides of the transversal are alternate interior. Two exterior angles that have di¤erent vertices and lie on opposite sides of the transversal are alternate exterior. Parallel Lines and Congruent Angles Now we are going to consider two parallel lines cut by a transversal . Postulate 11: kIf two parallel lines are cut by a transversal, then the corresponding angles are congruentk Example 1: If l km and m 6 1 = 110 . Find: 1 3 5 7 6 8 2 4 a) m6 2 b) m6 5 c) m6 4 d) m6 8 l m v Page: 3 Bibiana Lopez Elementary Geometry by Alexander and Koeberlein 2.1 Theorem 2.1.2: kIf two parallel lines are cut by a transversal, then the alternative interior angles are congruentk Example 2: (Proof of Theorem 2.1.2) Given: Prove: l km and transversal v 3=6 6 6 PROOF Statements Reasons 1 3 5 7 2 4 6 l m 8 v Theorem 2.1.3: kIf two parallel lines are cut by a transversal, then the alternative exterior angles are congruentk Parallel Lines and Supplementary Angles Theorem 2.1.4: kIf two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementaryk Example 3: (Proof of Theorem 2.1.4) ! ! ! Given: T V W Y with transversal RS Prove: 6 1 and 6 3 are supplementary PROOF Statements Reasons Page: 4 Bibiana Lopez Elementary Geometry by Alexander and Koeberlein 2.1 Theorem 2.1.5: kIf two parallel lines are cut by a transversal, then the exterior angles on the same side of the transversal are supplementaryk Example 4: ! ! ! Given: T V W Y with transversal RS m 6 RU V = (x + 4) (x m 6 W XS = x2 Find: 3) 3 x Example 5: (Exercise 12) Given: AD BC , AB DC m 6 A = 92 Find: a) m6 B A b) m6 C c) m6 D B C D WARNING! Do not assume properties from drawings that are not given. For example, from the drawing above it seems that angles are right angles (90 ) but they are not!!!! Page: 5 Bibiana Lopez Elementary Geometry by Alexander and Koeberlein 2.1 Example 6: (Exercise 17) Given: m kn and transversal k m 6 3 = 6x + y m 6 5 = 8x + 2y m 6 6 = 4x + 7y Find: x; y and m6 7 Page: 6 Bibiana Lopez