BELL RINGER What is the measure of ABC?

Chapter 3: Parallel and Perpendicular Lines Lesson 3.3: Proving Lines are Parallel

Standard/Objectives: Students will learn and apply geometric concepts Objectives: Prove that two lines are parallel. Use properties of parallel lines to solve real-life problems, such as proving that prehistoric mounds are parallel. Properties of parallel lines help you predict.

Postulate 16: Corresponding Angles Converse If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.

Theorem 3.4: Alternate Interior Angles Converse If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.

Theorem 3.5: Alternate Exterior Angles Converse If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel.

Theorem 3.6: Consecutive Interior Angles Converse If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel.

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

Example 1: Proof of Alternate Interior Converse Statements: 1.  1   2 2.  2   3 3.  1   3 4.m ║ n Reasons: 1. Given 2. Vertical Angles 3. Transitive prop. 4. Corresponding angles converse

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

Paragraph Proof You are given that  4 and  5 are supplementary. By the Linear Pair Postulate,  5 and  6 are also supplementary because they form a linear pair. By the Congruent Supplements Theorem, it follows that  4   6. Therefore, by the Alternate Interior Angles Converse, g and h are parallel.

Find the value of x that makes j ║ k. Solution: Lines j and k will be parallel if the marked angles are supplementary. x  + 4x  = 180  5x = 180  X = 36  4x = 144  So, if x = 36, then j ║ k. xx 4x 

Theorem 3.7: Transitive Property of Parallel Lines If two lines are parallel to the same line, then they are parallel to each other.

Example 5: Use the Transitive Property of Parallel Lines In the figure each rung of the ladder is parallel to the rung directly above it. Explain why the top rung is parallel to the bottom rung.

Conclusion: Two lines are cut by a transversal. How can you prove the lines are parallel? Show that either a pair of alternate interior angles, or a pair of corresponding angles, or a pair of alternate exterior angles is congruent, or show that a pair of consecutive interior angles is supplementary.

HOMEWORK ASSIGNMENT Page #1, 3-15, 17-23, 29-35