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.

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. D Exterior angle 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.

Given: exterior angle BCD Prove: M A C D 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:

(short form: Alt. int. lines.) 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.) Given: a Prove: a ll b 3 b 6 Use an indirect proof to prove that a ll b.

(short form: Alt. ext. lines.) 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.) Given: a Prove: a ll b 1 b 8 This can be proved by use of Alt. int. lines.

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

This can be proved by use of Alt. int. lines. 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. Given: a Prove: a ll b 4 b 6 This can be proved by use of Alt. int. lines.

This can be proved by use of Alt. int. lines. 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. Given: a 2 Prove: a ll b b 8 This can be proved by use of Alt. int. lines.

This can be proved by use of corr. lines. 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.

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

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

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

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