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. 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.

3. 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:

4. (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.

5. (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.

6. (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.

7. 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.

8. 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.

9. 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.

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

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

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

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