AMSCO'S Geometry. New York - Rye High School

Properties of Parallel Lines 337Theorem 9.3b If two parallel lines are cut by a transversal, then two interior angles on thesame side of the transversal are supplementary. (Converse of Theorem 9.3a)Given AB g CD gand transversal EF gEProve 4 is the supplement of 5.StrategyShow that 3 5 and that 4 is the supplementof 3. If two angles are congruent, then their supplementsare congruent. Therefore, 4 is also thesupplement of 5.A54 3BC DFThe proof of this theorem is left to the student. (See exercise 18.) SinceTheorems 9.2b and 9.3b are converses of Theorems 9.2a and 9.3a, we may statethe theorems in biconditional form:Theorem 9.2Two coplanar lines cut by a transversal are parallel if and only if correspondingangles are congruent.Theorem 9.3Two coplanar lines cut by a transversal are parallel if and only if interiorangles on the same side of the transversal are supplementary.EXAMPLE 1Transversal EF gintersects AB gand CD gat G and H,respectively. If AB g CD g,mBGH 3x 20, andmGHC 2x 10:AEGBa. Find the value of x. b. Find mGHC.c. Find mGHD.CHFDSolution a. Since AB g CD gand these lines are cut by transversal EF g, the alternate interiorangles are congruent: mBGH mGHC3x 20 2x 103x 2x 10 20x 30b. mGHC 2x 10 2(30) 10 70