We cannot construct a unique triangle, if we are given

CONCEPT: A triangle can be constructed as a UNIQUE triangle only when the given data fixes both its shape and its size. The standard construction criteria are:

• SSS: three sides

• SAS: two sides and the included angle

• ASA / AAS: two angles and a side

• RHS: right angle, hypotenuse and one side (for right triangles)

Each of these criteria pins down both the shape and the size of the triangle, so exactly one triangle satisfies them. Three angles alone (AAA), however, fix only the SHAPE of the triangle through similarity -- the SIZE is left completely free, because triangles of any size can share the same three angles.

APPLICATION: Checking each option against this rule -- two angles and one side is the ASA/AAS criterion (fixes shape and size); only three sides is the SSS criterion (fixes shape and size); two sides and the included angle is the SAS criterion (fixes shape and size). Only three angles is the AAA case, which fixes shape but leaves size undetermined, so no unique triangle can be constructed from it.

CROSS-CHECK: An equilateral triangle of side 2 cm and one of side 10 cm both have all three angles equal to 60 degrees, yet they are clearly different triangles. This confirms that knowing only the three angles can never single out one triangle.