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5.2 Proving That Lines Are Parallel

5.2 Proving That Lines Are Parallel. Objective: After studying this section, you will be able to apply the exterior angle inequality theorem and use various methods to prove lines are parallel.

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5.2 Proving That Lines Are Parallel

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  1. 5.2 Proving That Lines Are Parallel Objective: After studying this section, you will be able to apply the exterior angle inequality theorem and use various methods to prove lines are parallel.

  2. An exterior angle of a triangle is formed whenever a side of the triangle is extended to form an angle supplementary to the adjacent interior angle. Exterior angle D Adjacent Interior angle F E Remote Interior angle Theorem: The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.

  3. Given: exterior angle BCD B Prove: P M A D C Locate the midpoint, M, of BC. Draw AP so that AM=MP Draw CP Triangles AMB and PMC are congruent because of SAS This makes angle B and angle MCP congruent (CPCTC) This also proves that angle BCD is greater than angle B Extend BC, creating a vertical angle to angle BCD The following results:

  4. Theorem: If two lines are cut by a transversal such that two alternate interior angles are congruent, the lines are parallel. (short form: Alt. int. lines.) a Given: Prove: a ll b 3 b 6 Use an indirect proof to prove that a ll b.

  5. Theorem: If two lines are cut by a transversal such that two alternate exterior angles are congruent, the lines are parallel. (short form: Alt. ext. lines.) a Given: 1 Prove: a ll b b 8 This can be proved by use of Alt. int. lines.

  6. Theorem: If two lines are cut by a transversal such that two corresponding angles are congruent, the lines are parallel. (short form: corr. lines.) a Given: 2 Prove: a ll b b 6 This can be proved by use of Alt. int. lines.

  7. Theorem: If two lines are cut by a transversal such that two interior angles on the same side of the transversal are supplementary, the lines are parallel. a Given: Prove: a ll b 4 b 6 This can be proved by use of Alt. int. lines.

  8. Theorem: If two lines are cut by a transversal such that two exterior angles on the same side of the transversal are supplementary, the lines are parallel. a Given: 2 Prove: a ll b b 8 This can be proved by use of Alt. int. lines.

  9. Theorem: If two coplanar lines are perpendicular to a third line, they are parallel. a b Given: Prove: a ll b c This can be proved by use of corr. lines.

  10. a b Given: Prove: a ll b 1 2 c

  11. Given: Prove: MATH is a parallelogram A T 3 1 2 4 H M

  12. Given: Prove: FROM is a TRAPEZOID S 3 O R 1 2 F M

  13. Summary Name the different ways to prove lines are parallel. Homework: worksheet

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