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3-3 Proving Lines Parallel

3-3 Proving Lines Parallel. Objectives. Students will be able to Determine whether two lines are parallel Write flow proofs Define and apply the converse of the theorems from the previous section. Essential Understanding.

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3-3 Proving Lines Parallel

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  1. 3-3 Proving Lines Parallel

  2. Objectives • Students will be able to • Determine whether two lines are parallel • Write flow proofs • Define and apply the converse of the theorems from the previous section

  3. Essential Understanding • You can use certain angle pairs to determine if two lines are parallel

  4. What is the corresponding angles theorem? • If a transversal intersects two parallel lines, then corresponding angles are congruent • What is the converse of the corresponding angles theorem? • If two lines and a transversal form congruent corresponding angles, then the lines are parallel

  5. Identifying Parallel Lines • Which lines are parallel if <6 ≅ <7? • m || l • Which lines are parallel if <4 ≅ <6 • a || b

  6. Converse of the Alternate Interior Angles Theorem • If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel

  7. Converse of the Same-Side Interior Angles Theorem • If two lines and a transversal form same side interior angles that are supplementary, then the two lines are parallel.

  8. Converse of the Alternate Exterior Angles Theorem • If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel.

  9. Summary • If corresponding angles are congruent, then the lines are parallel • If alternate interior lines are congruent, then the lines are parallel • If alternate exterior lines are congruent, then the lines are parallel • If same side interior angles are supplementary, then the lines are parallel

  10. Things to Keep in Mind… • In order to use the theorems relating to parallel lines, you must first prove the lines are parallel if it is not given/stated in the problem. • Even if lines appear to be parallel, you cannot assume they are parallel • Always assume diagrams are NOT drawn to scale, unless otherwise stated

  11. Flow Proof • Third way to write a proof • In a flow proof, arrows show the flow, or the logical connections, between statements. • Reasons are written below the statements

  12. Proof of the Converse of Alternate Interior Angles Theorem • Given: <4 ≅ <6 • Prove: l || m <4 ≅ <6 Given <2 ≅ <6 L || m Trans. Prop of ≅ <2 ≅ <4 Converse of Corresponding Angles Thm. Vert. <s are ≅ *You cannot use the Corresponding Angles Thm to say <2 ≅ <6 because we do not know if the lines are parallel

  13. Write a flow proof • Given: m<5 = 40, m<2 = 140 • Prove: a || b • Start with what you know • The given statement • What you can concludefrom your picture. • What you need to know • Which theorem you can use to show a||b

  14. Write a flow proof • Given: m<5 = 40, m<2 = 140 • Prove: a || b <5 and <2 are Supp. <s <5 = 40 Given Def. of Supp. <s a || b <2 = 140 Converse of Same Side Int. <sThm Given <5 and <2 are Same side Interior Angles Def. of Same Side Interior <s

  15. You now have four ways to prove if two lines are parallel

  16. Using Algebra • What is the value of x for which a || b? • Work backwards. What must be true of the given angles for a and b to be parallel? • How are the angles related? • Same side interior • Therefore, they must add to be 180

  17. Using Algebra • What is the value of x for which a || b? • Work backwards. What must be true of the given angles for a and b to be parallel? • How are the angles related? • Corresponding Angles • Therefore, the angles are congruent

  18. Homework • Pg. 160 – 162 • # 7 – 16, 21 – 24, 28, 32 • 16 Problems

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