Chapter 5 Section 2: Proving That Lines Are Parallel An Epic PowerPoint Presentation by Marco Richione …with special guest: The Banana

The Exterior Angle Inequality Theorem Any exterior angle of a triangle is formed when a side of the triangle is extended to form an angle supplementary to the adjacent interior angle Theorem #30: “The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.” Gee whiz! Remote Interior Angle I  Remote Interior Angle II  Exterior Angle Adjacent Interior Angle 

Identifying Parallel Lines When two lines are cut by a transversal, eight angles are formed. You can use several pairs of these angles to prove the the lines are parallel. Theorem #31: “If two lines are cut by a transversal such that two alternate interior angles are congruent, the lines are parallel.” Shortcut Form for Theorem #31: “Alt. int. s => || lines” Given: 3 6 Prove: a || b a 3 6 b

Identifying Parallel Lines Cont’d… Theorem #32: “If two lines are cut by a transversal such that two alternate exterior angle are congruent, the lines are parallel. Shortcut Form for Theorem #32: “Alt. ext. s => || lines” 5 This theorem works since the supplement of the exterior angles are congruent, meaning the alternate interior angles are congruent. We learned in the previous side the theorem that proves || lines by alt. int. angles congruent. b c 9 Given: 5 9 Conclusion: b || c

Identifying Parallel Lines Cont’d… Theorem #33: “If two lines are cut by a transversal such that two corresponding angles are congruent, the lines are parallel.” Shortcut to Theorem #33: “Corr. s => || lines” 3 x 7 y Given: 3 7 Prove: x || y

Identifying Parallel Lines Cont’d… Theorem #34: “If two lines are cut by a transversal such that two interior angles on the same side of the transversal are supplementary, the lines are parallel.” Shortcut Form to Theorem #34: “Same side int. s supp. => || lines” c 1 9 d Given: 1 suppl. 9 Prove: c || d

Identifying Parallel Lines Cont’d… Theorem #35: “If two lines are cut by a transversal such that two exterior angles on the same side of the transversal are supplementary, the lines are parallel.” Shortcut Form to Theorem #35: “Same side ext. s suppl. => || lines” 3 g h 7 Given: 3 suppl. 7 Prove: g || h

Identifying Parallel Lines Cont’d… Theorem #36: “If two coplanar lines are perpendicular to the same third line, they are parallel.” d e f Given: df and ef Prove: d || e

Sample Problems I Difficulty Level: Easy 1 a Given: 1 4 Prove: a || b 4 b Solution Proof… _______________________________________ 1 4 1. Given 2. a || b 2. If two lines are cut by a transversal such that two corresponding angles are congruent, the lines are parallel.

Sample Problems II Difficulty Level: Moderate YOU CAN DO IT! A 3 B 1 4 2 D C Given: 1 2 DAB BCD Prove: ABCD is a parallelogram Solution Proof… ____________________________________ 1 2 1. Given 2. PQ || RS 2. Alt. int. s => || lines 3. DAB BCD 3. Given 4. 3 4 4. Subtraction Property 5. QR || PS 5. Same as 2 6. ABCD is a 6. 2 pairs opp. Side || => quad. is a parallelogram

Practice Problems a b 1.) 2.) Given: a &b lie in the same plane. b a 1 Part One: State which theorem learned in this section can sufficiently prove that lines a &b are parallel a b 1.) 2.) Given: a &b lie in the same plane. b a Part Two: Solve for the measure of the missing angle with the information given. 1 a Given: The following pairs of angles are congruent… 1 & 2; 3 & 4; 5 &6 The following pairs of angles are supplementary… 7 & 8; 9 & 10 3.) 2 3 5 b 7 6 4 c Objective: State all of the possible answers for which of the 6 lines are parallel from the given information. d 8 9 10 e f

Practice Problems: Solutions Part One: Solutions 1.) Theorem #33: “Corresponding Angles Congruent Imply Parallel Lines” can be used to solve this problem. Since the angles are marked congruent by the given mark, and the shape of the angles formed when the angles are traced resembles and F shape, the angles are corresponding angles. 2.) Theorem#36: “Coplanar Lines Perpendicular to the Same Line Imply Parallel Lines” can be used to solve this problem. Lines A and B are both perpendicular to the line between them, given randomly the name Line C. Since the square mark located at the interception point of Lines A and B with Line C imply that a right angle is formed, the lines are perpendicular to Line C since “Right Angles Imply Perpendicular Lines.” Thus, Lines A and B are coplanar and are both perpendicular to the same line, Line C. Therefore, Theorem #36 can be used to solve this problem. Part Two: Solutions 3.) By given that the following angles are congruent: a || b; b ||c By given that the following angles are supplementary: c || d; e || f *Note: By Transitive Property, more lines can be assumed to be parallel to one another. It can be assumed that since both Lines A and C are parallel to Line B, a || c. It can also be assumed that since a || c and c || d (by exterior supplementary angles), a || d. Thus, a || b || c || d: all lines are parallel. However, e and f are only parallel to one another.

Works Cited… Geometry for Enjoyment and Challenge. New Edition. Evanston, Illinois: McDougal Littell, 1991. "Geometry: Parallel Lines ." Math for Morons Like Us. 1998. ThinkQuest Team. 30 May 2008<http://library.thinkquest.org/20991/geo/parallel.html>.