AMSCO'S Geometry. New York - Rye High School

Line Reflections 217Corollary 6.1aUnder a line reflection, angle measure is preserved.Proof: We found that ABC ABC.Therefore, ABC ABC becausethe angles are corresponding parts of congruent triangles.Corollary 6.1bUnder a line reflection, collinearity is preserved.BDAProof: Let D be a point on AB whose image is D. Since distance is preserved:AD AD DB DB AB = ABCkA, D, and B are collinear, D is between A and B, and AD DB AB. By substitution,AD DB AB.If D were not on ArBr, AD DB AB because a straight line is theshortest distance between two points. But by substitution, ADDBAB.Therefore, A, D and B are collinear and D is between A and B.Corollary 6.1cUnder a line reflection, midpoint is preserved.BCkAM MProof: Let M be the midpoint of AC and M the image of M. Since distance ispreserved under a line reflection, AM AM, and MC MC.Since M is the midpoint of AC, AM MC and, by the substitution postulate,AM MC.Therefore, M is the midpoint of ArCr, that is, midpoint is preserved undera line reflection.We can summarize Theorem 6.1 and its corollaries in the followingstatement: Under a line reflection, distance, angle measure, collinearity, and midpointare preserved.We use r kas a symbol for the image under a reflection in line k. Forexample,r k(A) B means “The image of A under a reflection inline k is B.”r k(ABC) ABC means “The image of ABC under a reflectionin line k is ABC.”