1. Similarity Axiom

Let ABC and XYZ be triangles. Suppose that for s = 1 or s = -1, m(ABC) = s m(XYZ), and suppose there is a constant k >
0 such that d(XY) = k d(AB) and d(YZ) = k d(BC). Then it follows that m(BCA) = s m(YZX), m(CAB) = s m(ZXY), and
d(XZ) = k d(AC).

Definitions

The triangles ABC and XYZ are similar (with respect to the orderings (A,B,C), (X,Y,Z) of their vertices) if there are
constants s and k with s = 1 or s = -1, and k > 0, such that the following six conditions hold:
m(ABC) = s m(XYZ)
m(BCA) = s m(YZX)
m(CAB) = s m(ZXY)
d(XY) = k d(AB)
d(YZ) = k d(BC)
d(XZ) = k d(AC)

The triangles ABC and XYZ are congruent (with respect to the orderings (A,B,C), (X,Y,Z) of their vertices) if there's a
constant s with s = 1 or s = -1 such that the following six conditions hold:
m(ABC) = s m(XYZ)
m(BCA) = s m(YZX)
m(CAB) = s m(ZXY)
d(XY) = d(AB)
d(YZ) = d(BC)
d(XZ) = d(AC)

2. Angle-Angle Similarity

Let ABC and XYZ be triangles. If there's a constant s such that s = 1 or s = -1, with m(CAB) = s m(ZXY) and m(ABC) =
s m(XYZ), then the triangles ABC and XYZ are similar.

3.Side-Side-Side Similarity

If ABC and XYZ are triangles, and there is a constant k > 0 such that d(XY) = k d(AB), d(YZ) = k d(BC), and d(ZX) = k
d(CA), then ABC and XYZ are similar.

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