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Lesson 3-5 Proving Lines Parallel • Postulate 3.4- If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel. Example: • Postulate 3.5- Parallel Postulate If a given line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line. Proving Lines Parallel Theorems 3.5 If two lines in a plane are cut by a transversal so that a pair of alternate exterior angles is congruent, then the two lines are parallel 3.6 If two lines in a plane are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the lines are parallel. 3.7 If two lines in a plane are cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel 3.8 In a plane, if two lines are perpendicular to the same line, then they are parallel Examples Determine which lines, if any, are parallel. supplementary. So, consecutive interior angles are consecutive interior angles are not supplementary. So, c is not parallel to a or b. Answer: Determine which lines, if any, are parallel. Answer: ALGEBRA Find x and mZYN so that Explore From the figure, you know that and You also know that are alternate exterior angles. Plan For line PQ to be parallel to MN, the alternate exterior angles must be congruent. Substitute the given angle measures into this equation and solve for x. Once you know the value of x, use substitution to find Solve Alternate exterior angles Substitution Subtract 7x from each side. Add 25 to each side. Divide each side by 4. Original equation Simplify. Examine Verify the angle measure by using the value of x to find Since Answer: ALGEBRA Find x and mGBA so that Answer: Given: Prove: Proof: Statements 1. 2. 3. 4. 5. 6. 7. . . Reasons 1. Given 2. Consecutive Interior Thm. 3. Def. of suppl. s 4. Def. of congruent s 5. Substitution 6. Def. of suppl. s 7. If cons. int. s are suppl., then lines are . Given: Prove: Proof: Statements Reasons 1. 1. Given 2. 2. Alternate Interior Angles 3. 3. Substitution 4. . 4. Definition of suppl. s 5. 5. Definition of suppl. s 6. 6. Substitution 7. 7. If cons. int. s are suppl., then lines are . Answer: Answer: Since the slopes are not equal, r is not parallel to s.