Which of the following were the two types of constructions that were never accomplished by the Greeks using only a compass and straightedge?

Actually there are three types of construction that were never accomplished by Greeks using compass and straightedge these are squaring a circle, doubling a cube and trisecting any angle.

The problem of squaring a circle takes on unlike meanings reliant on how one approaches the solution. Beginning with Greeks Many geometric approaches were devised, however none of these methods accomplished the task at hand by means of the plane methods requiring only straightedge and a compass.

The origin of the problem of doubling a cube also referred as duplicating a cube is not certain. Two stories have come down from the Greeks regarding the roots of this problem. The first is that the oracle at Delos ordered that the altar in the temple be doubled over in order to save the Delians from a plague the other one relates that king Minos ordered that a tomb be erected for his son Glaucus.

The structure of regular polygons and the structure of regular solids was a traditional problem in Greek geometry. Cutting an angle into identical thirds or trisection was another matter overall. This was necessary to concept other regular polygons. Hence, trisection of an angle became an significant problem in Greek geometry.