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Geo 3.4 proving lines parallel

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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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Page 2: Geo 3.4 proving lines 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.

Page 3: Geo 3.4 proving lines parallel

Biconditional:

• Two lines cut by a transversal are parallel if and only if they the corresponding angles are congruent.

Page 4: Geo 3.4 proving lines parallel

• 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.

Page 5: Geo 3.4 proving lines 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.

Page 6: Geo 3.4 proving lines 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.

Page 7: Geo 3.4 proving lines 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

//

Page 8: Geo 3.4 proving lines parallel

Sailing

If two boats sail at an angle of 45o to the wind and the wind is constant, will their paths ever cross?

Page 9: Geo 3.4 proving lines parallel

Solution

• Because corresponding angles are congruent, the boats’ paths are parallel.

• Parallel lines do not intersect, so the boats’ paths will not cross.

Page 10: Geo 3.4 proving lines parallel

Practice

• TB p153: 10-28 even, 44-48 (all)

Maybe you could be a bit more explicit at step 3.