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Section shows how to prove two lines to be parallel using the converses of the postulates and theorems of previous lessons
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Proving Lines are Parallel
Properties of Parallel LinesCorresponding Angles Postulate:
• If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
Converse:• If two lines are cut by a transversal
so that corresponding angles are congruent, then the lines are parallel.
Biconditional:
• Two lines cut by a transversal are parallel if and only if they the corresponding angles are congruent.
• If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
Theorem: Alternate Interior Angles:
Converse:
• If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.
• If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.
Theorem: Consecutive Interior Angles:
Converse:
• If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel.
• If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.
Theorem: Alternate Exterior Angles:
Converse:
• If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel.
Proof of Alternate Interior Angles Converse
Statement Reason
1 ∠1 ≅ ∠2 Given
2 ∠2 ≅ ∠3 Vertical angles theorem
3 ∠1 ≅ ∠3Transitive property of congruence
4 l // mConverse of corresponding angles postulate
//
Sailing
If two boats sail at an angle of 45o to the wind and the wind is constant, will their paths ever cross?
Solution
• Because corresponding angles are congruent, the boats’ paths are parallel.
• Parallel lines do not intersect, so the boats’ paths will not cross.
Practice
• TB p153: 10-28 even, 44-48 (all)
Maybe you could be a bit more explicit at step 3.