1. 9-3: The Parallel Postulate 9-4: Triangles: Quick Revisit
Proof Geometry

2. Alternate Interior Parallel Converse
What did Alternate Interior Parallel Theorem say, again? If alternate interior angles are congruent then lines are parallel. Therefore its converse says…

3. Alternate Interior Parallel Converse
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
then

4. Alt. Int. Parallel Converse Proof
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
Given: Parallel lines L1 and L2 with transversal T intersecting them at P and Q
Suppose: a and b are not congruent(supposition)
Then: Let L be the line through P for which alternate interior angles are congruent. Then L || L2.
But: It is given that L1||L2. The Parallel Postulate assures only one parallel line through external point.
(the CONTRADICTION)
So: a  b

5. The Corresponding Angles Parallel Converse
What did Corresponding Angles Parallel say, again? If corresponding angles are congruent then lines are parallel. Therefore its converse says…

6. The Corresponding Angles Parallel Converse
If two parallel lines are cut by a transversal, then corresponding angles are congruent.
then

7. Same Side Angle Parallel Converse
What did Same Side Angle Parallel say, again? If same side interior angles are supplementary then lines are parallel. Therefore…

8. Same Side Angle Parallel Converse
If two parallel lines are cut by a transversal, then same side interior angles are supplementary. then m2 + m3 = 180 and m1 + m4 = 180

9. Revisit Triangle Sum Theorems
The sum of the interior angles in a triangle is 180˚.

10. Triangle Fundamentals
Proof of triangle sum theorem

11. Third Angles Theorem
Third Angles Theorem:

12. Homework
pg : #5 – 11 odd, 16 pg : #2, 5, 7, 8