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Parallel Lines cut by a Transversal Geometry Standard 7.0 Prove & use theorems involving the properties of parallel lines cut by a transversal. Parallel Lines Cut by a Transversal m j k Lines j & k are parallel. They are cut by the transversal line m. Parallel lines ? Parallel Lines? Parallel Lines Cut by Parallel Lines Parallel Fields Parallel Walls Parallel Church Parallel City Parallel π’s? Parallel Lines in a Quilt Parallel Lines Cut by a Transversal Behavior of coplanar lines and slopes • Intersecting lines have different slopes. 4 2 -5 5 -2 -4 -6 • Parallel lines have the same slope. 4 2 -5 5 -2 -4 -6 Huh? What this comes down to is the following: If angles 1 and 2 have measures that add up to less than 180 degrees, then the non-transversal lines forming them will intersect. 1 2 So, the Parallel Postulate states the converse How many Parallel lines do you see? How many transversals? Are stair steps always parallel? Parallel, Skew or…? Which are the Parallel Lines? Which are the Transversals? Things are getting a little “stair” crazy! And more so!!! So, in the terms of high school geometry: Given a line & a point not on the line, there is exactly one line in the plane of the given line, that goes through the point parallel to the line. So, on a plane, when parallel lines are crossed by a transversal, what happens? m 1 j 2 4 3 5 k 6 8 7 Names of Associated Angles m j 2 1 3 4 5 6 7 8 k • Alternate Exterior Angles are on opposite sides of the transversal – 1 & 7 – 2 & 8 •Corresponding Angles lie on the same side of the transversal situated the same way on two different parallel lines. (Angles 2 & 6, 3 & 7, 1 & 5, 4 & 8 • Interior Angles lie between the two lines – 3, 4 ,5, 6 • Alternate Interior Angles are on opposite sides of the transversal – – 3 &5 4 &6 • Consecutive Interior Angles are on the same side of the transversal – 3 &6 – 4 &5 • Exterior Angles lie outside the two lines – 1, 2, 7, 8 What do we know? m 1 j k 2 3 4 5 6 7 8 • Vertical angles are congruent. – 1 & 3 – 2 & 4 – 5 & 7 – 6 & 8 • Linear pairs form supplementary angles. – 1 & 2 – 2 & 3 – 3 & 4 – 4 & 1 – 5 & 6 – 6 & 7 – 7 & 8 – 8 & 1 When two parallel lines are cut by a transversal, the following statements are true: • • • • Corresponding angles are congruent. Alternate interior angles are congruent. Alternate exterior angles are congruent. Interior angles on the same side of the transversal are supplementary. • If a transversal is perpendicular to one of the two parallel lines, then it is automatically parallel to the other line m j k 1 2 3 4 6 5 7 8 Theorems or Postulates? • Choose one of the properties on the last slide to be a postulate & then you can prove all the others by applying it. – Postulate: If two parallel lines are cut by a transversal, then corresponding angles are congruent. m 1 j k 2 3 4 5 6 7 8 1 R5 R2 R6 R4 R8 R3 R7 Alternate Interior Angle Theorem: It two parallel lines are cut by a transversal, then alternate interior angles are congruent. m • Given: j || m • Prove: 3 5 & 4 6 2 1 j k 4 3 5 6 7 8 Statements 1. j || m 2. 1 R5, R2 R6, R4 R8, R3 R7 3. 1 R3, R5 R7 R6 R8, R2 R4 4. 3 R5, R4 R6 Reasons 1. 2. 3. 4. Alternate exterior Theorem: If two parallel lines are cut by a transversal then the alternating exterior angles are congruent. m 1 3 q 2 4 Given: m || n, q is a transversal; Prove: 2 R7, R1 R8 5 6 7 8 We have proven that when parallel lines are cut by a transversal, alternate interior angles are congruent. So we know that angles 3 & 6 & angles 4 & 5 are congruent. Since angles 2 & 3, angles 1 & 4, angles 5 & 8 & angle 6 & 7 are vertical angles, they are also congruent. So by the transitive property applied twice, angles 2 & 7 & angles 1 & 8 are congruent. n Consecutive Angles Theorem: If two parallel lines are cut by a transversal then the consecutive interior angles formed are supplementary