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Parallel Lines and Transversals Angle pairs formed by parallel lines and a transversal are congruent, supplementary, or both. t 12 43 56 87 l m If two parallel lines are cut by a transversal, then alternate interior angles are t congruent. If l || m, then 4 6 and 3 5. 12 43 56 87 l m If two parallel lines are cut by a transversal, then alternate exterior angles are t congruent. If l || m, then 1 7 and 2 8. 12 43 56 87 l m If two parallel lines are cut by a transversal, then corresponding angles are congruent. t If l || m, then 1 5, 2 6, 4 8, and 3 7. 12 43 56 87 l m If two parallel lines are cut by a transversal, then same side interior angles t (consecutive interior angles) are supplementary. If l || m, then 4 and 5 are supplementary, 3 and 6 are supplementary. 12 43 56 87 l m Are any other angle pairs congruent? t 12 43 56 87 l m Are any other angle pairs congruent? Vertical angle pairs are congruent. t 12 43 56 87 l m Are any other angle pairs supplementary? t 12 43 56 87 l m Are any other angle pairs supplementary? Angles that form a linear pair are supplementary. t 12 43 56 87 l m There are only two angle measures formed when two parallel lines are cut by a transversal. t 12 43 56 87 l m If l || m, and m1 = 110°, find the remainder of the angle measures. t 12 43 56 87 l m If l || m, and m1 = 110°, find the remainder of the angle measures. t 110° 70° 12 70° 4 3 110° 110° 5 6 70° 8 7110° 70° l m Proving Lines Parallel Angle pairs that are congruent or supplementary can be used to determine whether two lines are parallel. t 12 43 56 87 l m If two lines are cut by a transversal and alternate interior angles are congruent, t then the lines are parallel. If 4 6 or 3 5, then l || m. 12 43 56 87 l m If two lines are cut by a transversal and alternate exterior angles are congruent, t then the lines are parallel. If 1 7 or 2 8, then l || m. 12 43 56 87 l m If two lines are cut by a transversal and corresponding angles are congruent, then t the lines are parallel. If 1 5, 2 6, 4 8, or 3 7, then l || m. 12 43 56 87 l m If two lines are cut by a transversal and same side interior angles (consecutive t interior angles) are supplementary, then the lines are parallel. If 4 and 5 are supplementary or 3 and 6 are supplementary, then l || m. 12 43 56 87 l m Parallel and Perpendicular Lines Use the relationship of two lines to a third line to determine whether the two lines are parallel or perpendicular. If two lines are parallel to the same line, then they are parallel to each other. If a || c and b || c, then a || b. a b c In a plane, if two lines are perpendicular to the same line, then they are parallel. t If m t and n t, then m || n. m n In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other. If n l and l \\ m, then n m. Parallel Lines and Triangles Through a point not on a line, there is one and only one line parallel to the given line. P Through a point not on a line, there is one and only one line parallel to the given line. P Sum of the Interior Angles of a Triangle Lets use this concept to explore the angles of a triangle. Given ABC , construct a line through point B, parallel to AC. B A C Given ABC , construct a line through point B, parallel to AC. B A C What do we know about m1, m2, and m3? B 1 A 2 3 C What do we know about m1, m2, and m3? m1 + m2 + m3 = 180°, these angles together form a straight angle, straight angles measure 180. B 1 A 2 3 C What do we know about m1 and mA, m3 and mC? B 1 A 2 3 C What do we know about m1 and mA, m3 and mC? m1 = mA, m3 = mC, If two parallel lines are cut by a transversal, alternate interior angles are congruent. B 1 A 2 3 C If m1 + m2 + m3 = 180°, and m1 = mA, m3 = mC, by substitution, mA + m2 + mC = 180°. B 1 A 2 3 C This means the sum of the interior angles of a triangle is 180°. B 1 A 2 3 C Exterior Angle of a Triangle An exterior angle of a triangle is formed by extending a side of the triangle. exterior angle For each exterior angle, the two nonadjacent interior angles are called its remote interior angles. remote interior angle exterior angle remote interior angle Exterior Angle Theorem The measure of each exterior angle of a triangle equals the sum of its two remote interior angles. m4 = m1 + m2 remote interior angle 1 2 remote interior angle 3 exterior angle 4