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Proving Lines Parallel: Theorems, Proofs, Examples

Learn how to prove lines parallel using Corresponding Angles Theorem, Alternate Interior Angles Theorem, Same-Side Interior Angles Postulate, and more in this instructional guide. Includes step-by-step proofs and examples. Homework assignments provided.

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Proving Lines Parallel: Theorems, Proofs, Examples

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  1. Section 3-3Proving Lines Parallel –Day 1, Calculations. Michael Schuetz

  2. Theorem 3-4: Converse of the Corresponding Angles Theorem. l If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel. 2 m 6

  3. Theorem 3-5: Converse of the Alternate Interior Angles Theorem. l If two lines and a transversal form alternate interior angles that are congruent, then the lines are parallel. 4 m 6

  4. Theorem 3-6: Converse of the Same-Side Interior Angles Postulate. l If two lines and a transversal form same-side interior angles that are supplementary, then the lines are parallel. 3 m 6

  5. Theorem 3-7: Converse of the Alternate Exterior Angles Theorem. l If two lines and a transversal form alternate exterior angles that are congruent, then the lines are parallel. 1 m 7

  6. Example 1, Identifying parallel lines • Which lines are parallel if and ? Justifies your answers? p q Converse of the corresponding angles theorem Converse of the alternate interior angles theorem l 2 5 8 1 3 m 9 4 6 7

  7. Example 2, Using Algebra • What is the value of x that makes p//q? Which theorem or postulate justifies your answer? The converse of the Same-Side Interior Postulate tells us that to make p//q, then p q l 2x+9˚ 111˚

  8. Homework: Day 1 • P. 160, #’s 7, 8, 15, 16, 27, 28, 31, 32, 47, 48

  9. Section 3-3Proving Lines Parallel –Day 2, Proofs. Michael Schuetz

  10. Proof of Theorem 3-5: l 2 • Given: • Prove: l//m 4 Statements Reasons m 6 1. Given 2. Vertical angles are congruent 3. Transitive property 4. Theorem 3-4: If corresponding angles are congruent then the lines are parallel.

  11. Proof of Theorem 3-7: l 1 • Given: • Prove: l//m 3 m Statements Reasons 1. Given 7 2. Vertical angles are congruent 3. Transitive property 4. Theorem 3-4: If corresponding angles are congruent then the lines are parallel.

  12. Proof of Theorem 3-4: l 1 • Given: • Prove: l//m 4 m Statements Reasons 5 1. Given 2. Angles 1 and 4 form a linear pair 3. Substitution 4. Theorem 3-6: If same-side interior angles are supplementary then the lines are parallel.

  13. Homework: Day 2 • P. 161, #’s 17-26, for 40 & 41 write a 2 column proof.

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