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Do Now: B A Given: C -XB bisects AXC -angle EXC is a straight angle. X E 75 ° Find (not prove): D -Find measure of angle BXC and angle BXA Geometry & A.K.A.: How can we show that two lines are parallel? 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). A B C D Notation: AB || CD Line AB is parallel to line CD 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) 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. What can we say about the angles formed by a transversal? Exterior Interior What are the exterior angles? 1, 2, 7, and 8 What are the interior angles? 3, 4, 5, and 6 2 1 3 4 6 5 7 8 Alternate Interior Angles Interior angles on OPPOSITE sides of the transversal at different vertices. The alternate interior angles are: 4 and 6 3 and 5 THESE COME IN PAIRS! 4 and 5 are NOT alternate interior angles. 2 1 3 4 6 5 7 8 Corresponding Angles Angles in the same position, but at different vertices along the transversal. Examples: 1 and 5 2 and 6 3 and 7 5 and 6 are NOT corresponding angles 2 1 3 4 6 5 7 8 Do Now Identify at least one pair of angles that are a) Corresponding angles b) Alternate Interior Angles c) Supplementary angles d) Vertical angles 2 1 3 4 6 5 7 8 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. 2 1 3 4 6 7 5 8 This is how we show on a diagram that two lines are parallel What angles are congruent to angle 5 here? 7 (Vertical Angle) 3 (Alternate Interior Angle) 1 (Corresponding Angle) 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. Theorem IF A TRANSVERSAL CROSSES TWO PARALLEL LINES… • The interior angles on the same side of the transversal are supplementary. • The exterior angles on the same side of a transversal are supplementary. These theorems tell us that… These angle pairs are supplementary: 4&5 3&6 1&8 2&7 2 1 3 4 6 5 7 8