3.5 Proving Lines Parallel

Objectives Recognize angle conditions that occur with parallel lines Prove that two lines are parallel based on given angle relationships

Postulate 3.4 Converse of the Corresponding Angles Postulate If two lines in a plane are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. Abbreviation: If corr.  s are , then lines are ║.

Postulate 3.5 Parallel Postulate If given a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line.

Theorem 3.5 Converse of the Alternate Exterior Angles Theorem 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. Abbreviation: If alt ext.  s are , then lines are ║.

Theorem 3.6 Converse of the Consecutive Interior Angles Theorem 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. Abbreviation: If cons. int.  s are supp., then lines are ║.

Proof of the Converse of the Consecutive Interior Angles Theorem Given:  4 and  5 are supplementary Prove: g ║ h 6 g h 5 4

Paragraph Proof of the Converse of the Consecutive Interior Angles Theorem You are given that  4 and  5 are supplementary. By the Supplement Theorem,  5 and  6 are also supplementary because they form a linear pair. If 2  s are supplementary to the same , then  4   6. Therefore, by the Converse of the Corresponding  s Angles Postulate, g and h are parallel.

Theorem 3.7 Converse of the Alternate Interior Angles Theorem 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. Abbreviation: If alt. int.  s are , then lines are ║.

Proof of the Converse of the Alternate Interior Angles Theorem Given:  1   2 Prove: m ║ n m n

Two - Column Proof of the Converse of the Alternate Interior Angles Theorem Statements: 1.  1   2 2.  2   3 3.  1   3 4. m ║ n Reasons: 1. Given 2. Vertical Angles are  3. Transitive prop. 4. If corres.  s are , then lines are ║

Theorem 3.8 In a plane, if two lines are perpendicular to the same line, then they are parallel. Abbreviation: If 2 lines are ┴ to the same line, then the lines are ║.

Determine which lines, if any, are parallel. consecutive interior angles are supplementary. So, consecutive interior angles are not supplementary. So, c is not parallel to a or b. Answer: Example 1:

Determine which lines, if any, are parallel. Answer: Your Turn:

ALGEBRA Find x and m  ZYN so that Explore From the figure, you know that and You also know that are alternate exterior angles. Example 2:

Alternate exterior angles Subtract 7x from each side. Substitution Add 25 to each side. Divide each side by 4. Solve 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 Example 2:

Answer: Original equation Simplify. Examine Verify the angle measure by using the value of x to find Since Example 2:

ALGEBRA Find x and m  GBA so that Answer: Your Turn:

Given: Prove: Example 3:

Proof: 1. Given 1. ReasonsStatements 2. Consecutive Interior Thm Def. of congruent s If cons. int. s are suppl., then lines are. 7. Example 3: 4.  7 +  6 = Def. Suppl.  s 5.  4 +  6 = Substitution 6.  4 and  6 are suppl6. Def. Suppl.  s ;

Given: Prove: x || y a || b  1   12 Your Turn:

1. Given 1. a || b;  1   Corres.  s Postulate2.  1   Vertical  s are  3.  13   Substitution 4.  12   If corres.  s are , then lines are || 5. x || y Reasons Statements Proof: Your Turn:

Answer: Example 4:

Answer: Since the slopes are not equal, r is not parallel to s. Your Turn:

Assignment Geometry: Pg. 155 #13 – 31, 34, 35 Pre-AP Geometry: Pg. 155 #13 – 31,