Angles and Parallel Lines
Three ways in which two lines may be situated in space They may fail to intersect and fail to be coplanar: L1 and L3 Called skew They may intersect in a point: L1 and L2 In which case they are co-planar (Given two intersecting lines, there is exactly one plane containing both.) They may be coplanar without intersecting each other: L2 and L3 Called parallel
Definitions: Skew and Parallel Two lines are parallel if: They are coplanar They do not intersect We write On sketch
Definition: Transversal A transversal of two lines is a line which intersects them in two different points. Transversal Not Transversal
Transversal Transversals are often studied in relationship to parallel lines When a transversal intersects two lines eight angles are formed: 1 2 3 4 5 6 7 8
Classification of angles Corresponding Angles: Angles that occupy corresponding positions. 1 2 3 4 5 6 7 8
Classification of angles Corresponding Angles: Angles that occupy corresponding positions. Alternate: Angles that are on opposite sides of the transversal. 1 2 3 4 5 6 7 8
Classification of angles Corresponding Angles: Angles that occupy corresponding positions. Alternate: Angles that are on opposite sides of the transversal. Same Side Angles: Angles that are on the same sides of the transversal. 1 2 3 4 5 6 7 8
Classification of angles Corresponding Angles: Angles that occupy corresponding positions. Alternate: Angles that are on opposite sides of the transversal. Same Side Angles: Angles that are on the same sides of the transversal. Interior: Angles that are between the parallel lines. 3 4 5 6
Classification of angles Corresponding Angles: Angles that occupy corresponding positions. Alternate: Angles that are on opposite sides of the transversal. Same Side Angles: Angles that are on the same sides of the transversal. Interior: Angles that are between the parallel lines. 1 2 Exterior: Angles that are outside the parallel lines. 7 8
F Corresponding Angles 2 6, 1 5, 3 7, 4 8 Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then the Corresponding angles are congruent. 2 6, 1 5, 3 7, 4 8 F 1 2 3 4 5 6 7 8
Alternate Angles Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the Alternate Interior Angles are congruent. 3 6, 4 5 N Z 1 2 3 4 5 6 7 8
Alternate Angles Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then the Alternate Exterior Angles are congruent. 2 7, 1 8 1 2 3 4 5 6 7 8
Same Side or Consecutive Angles Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then the Consecutive Interior Angles are supplementary. m3 +m5 = 180º, m4 +m6 = 180º U 1 2 3 4 5 6 7 8
Lesson 2-4: Angles and Parallel Lines Example Lesson 2-4: Angles and Parallel Lines
Lesson 2-4: Angles and Parallel Lines Homework HW p 157 #10 – 16 even, 23, 27 – 30 all Lesson 2-4: Angles and Parallel Lines