Proving lines are parallel

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Proving lines are parallel

  1. 1. Proving Lines are Parallel
  2. 2. Properties of Parallel Lines <ul><li>Corresponding Angles Postulate: </li></ul><ul><li>If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. </li></ul>Converse: <ul><li>If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. </li></ul>
  3. 3. Biconditional: <ul><li>Two lines cut by a transversal are parallel if and only if they the corresponding angles are congruent. </li></ul>
  4. 4. <ul><li>If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. </li></ul>Theorem: Alternate Interior Angles: Converse: <ul><li>If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. </li></ul>
  5. 5. <ul><li>If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. </li></ul>Theorem: Consecutive Interior Angles: Converse: <ul><li>If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel. </li></ul>
  6. 6. <ul><li>If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. </li></ul>Theorem: Alternate Exterior Angles: Converse: <ul><li>If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. </li></ul>
  7. 7. Proof of Alternate Interior Angles Converse Statement Reason 1 ∠ 1 ≅ ∠ 2 Given 2 ∠ 2 ≅ ∠ 3 Vertical angles theorem 3 ∠ 1 ≅ ∠ 3 Transitive property of congruence 4 l ⊥ m Converse of corresponding angles postulate
  8. 8. Sailing <ul><li>If two boats sail at an angle of 45 o to the wind and the wind is constant, will their paths ever cross? </li></ul>
  9. 9. Solution <ul><li>Because corresponding angles are congruent, the boats’ paths are parallel. </li></ul><ul><li>Parallel lines do not intersect, so the boats’ paths will not cross. </li></ul>
  10. 10. Homework <ul><li>Exercise 3.4 page 153: 1-37, odd. </li></ul>
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