By Julian Lowell Coolidge

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We shall assume hereafter that every congruent transformation with which we deal has been enlarged to the greatest Under these circumstances possible extent. : Theorem 13. If two distinct points be invariant under a is true of all points of congruent transformation, the same their line. Theorem 14. If three non-collinear points be invariant * The idea of enlarging a congruent transformation to include additional points is due to Pasch, loc. cit. He merely assumes that if any point be adjoined to the one set, a corresponding point may be adjoined to the other.

AD belong to the internal %-ACB, AD must contain point E of CAB, and if we take P within (AE), once more If a \ 4- AGP < 4_ACD. Theorem 2. If, in any triangle, one side and an adjacent angle remain fixed, while the other side including this angle may be diminished at will, then the external angle opposite to the fixed side will take and retain a value differing from that of the fixed angle by less than any assigned value. Let the fixed side be (AB), while G is the variable vertex within a fixed segment (BD).

Is greater than that opposite 30. One side of a triangle cannot be greater than of the other two. Theorem the sum Tfieorem 31. The difference between two sides of a triangle than the third side. The proofs of these theorems are left to the reader. is less c2 CONGRUENT TRANSFORMATIONS 36 Theorem Two 32. CH; distinct lines cannot be coplanar with a third, and perpendicular to it at the same point. and perpendicular to Suppose, in fact, that we have AB* so that assume at A. by I. 31 may either of (CB) or of (CB').