AMSCO'S Geometry. New York - Rye High School

Basic Constructions 199Construction 4Bisect a Given Angle.GivenConstructBDE1. With B as centerand any convenientradius,draw an arc thatintersectsD and BC at E.Conclusionh BA hABCBF h, the bisector of ABCAADBBECC2. With D as center,draw an arc inthe interior ofABC.atBF h bisects ABC; ABF FBC.DEACF3. Using the sameradius, and with Eas center, draw anarc that intersectsthe arc drawn instep 2. Label thisintersection F.BDE4. Draw BF h.ACFProof We used congruent radii to draw BD > BE andDF > EF. By the reflexive property of congruence,BF > BF so by SSS, FBD FBE. Therefore,ABF FBC because they are correspondingparts of congruent triangles. Then BF hbisects ABCbecause an angle bisector separates an angle intotwo congruent angles.BDEAFCConstruction 5 will be similar to Construction 4. In Construction 4, anygiven angle is bisected. In Construction 5, APB is a straight angle that isbisected by the construction. Therefore, APE and BPE are right angles andPE g⊥ AB g.