AMSCO'S Geometry. New York - Rye High School

148 Congruence of Line Segments,Angles, and TrianglesCBDAFGProofHECongruent angles have the same measure. If ABD EFH, we can representthe measure of each angle by the same variable: mABD mEFH x.Since CBD is the complement of ABD, and GFH is the complementof EFH, then mCBD 90 x and mGFH 90 x. Therefore,mCBD mGFH and CBD GHF.Theorem 4.5If two angles are supplements of the same angle, then they are congruent.GivenProveABD is the supplement of DBC, and EBC is thesupplement of DBC.ABD EBCCEBDATheorem 4.6If two angles are congruent, then their supplements are congruent.GivenProveABD EFH, CBD is thesupplement of ABD, and GFH isthe supplement of EFH.CBD GFHCBDAGFHEThe proofs of Theorems 4.5 and 4.6 are similar to the proofs of Theorems4.3 and 4.4 and will be left to the student. (See exercises 18 and 19.)More Definitions and TheoremsInvolving Pairs of AnglesDEFINITIONA linear pair of angles are two adjacent angles whose sum is a straight angle.DCBAgIn the figure, ABD is a straight angle and C is not on ABD. Therefore,ABC + CBD ABD. Note that ABC and CBD are adjacent angleswhose common side is BC hand whose remaining sides are opposite rays thattogether form a straight line, AD g.Theorem 4.7If two angles form a linear pair, then they are supplementary.GivenProveABC and CBD form a linear pair.ABC and CBD are supplementary.DCBA