1. Do Now: B C D A E 75° Given: -XB bisects AXC -angle EXC is a straight angle. Find (not prove): -Find measure of angle BXC and angle BXA X

2. Geometry & A.K.A.: How can we show that two lines are parallel?

3. What are Parallel Lines? Definition: Parallel lines are coplanar (in the same plane) and either have NO points in common or every point in common (like the lines are on top of one another). Notation: AB C D AB || CDLine AB is parallel to line CD

4. Postulates relating to Parallel Lines Postulate: A line is parallel to itself (reflexive property) Postulate: If two lines are each parallel to the same line, they are parallel to each other. (transitive property)

5. What is a transversal? Definition: A transversal is a line that intersects two other lines in two DIFFERENT POINTS. This IS a transversal. This is NOT a transversal.

6. What can we say about the angles formed by a transversal? 1 2 4 3 5 6 7 8 Interior Exterior What are the exterior angles? 1, 2, 7, and 8 What are the interior angles? 3, 4, 5, and 6

7. Alternate Interior Angles 1 2 4 3 5 6 7 8 The alternate interior angles are: 4 and 6 3 and 5 THESE COME IN PAIRS! 4 and 5 are NOT alternate interior angles. Interior angles on OPPOSITE sides of the transversal at different vertices.

8. Corresponding Angles Angles in the same position, but at different vertices along the transversal. 1 2 4 3 5 6 7 8 Examples: 1 and 5 2 and 6 3 and 7 5 and 6 are NOT corresponding angles

9. Do Now 1 2 4 3 5 6 7 8 Identify at least one pair of angles that are a)Corresponding angles b)Alternate Interior Angles c)Supplementary angles d)Vertical angles

10. Aim: What is important about Corresponding and Alternate Interior angles? Theorem: If a transversal cuts (crosses) two parallel lines, the alternate interior angles are congruent. Theorem: If a transversal cuts two parallel lines, then corresponding angles are congruent.

11. This is how we show on a diagram that two lines are parallel 1 2 43 56 78 What angles are congruent to angle 5 here? 7 (Vertical Angle) 3 (Alternate Interior Angle) 1 (Corresponding Angle)

12. The Converse of both these theorems are true, too! Theorem: If a transversal crosses two lines, and the alternate interior angles are congruent, then the line are parallel. Theorem: If a transversal crosses two lines, and the corresponding angles are congruent, then the line are parallel.

13. Theorem The interior angles on the same side of the transversal are supplementary. The exterior angles on the same side of a transversal are supplementary. IF A TRANSVERSAL CROSSES TWO PARALLEL LINES…

14. These theorems tell us that… 1 2 4 3 5 6 7 8 These angle pairs are supplementary: 4 & 5 3 & 6 1 & 8 2 & 7