I divided a square into two parts producing a kite and dart and includes angles of 30, 60, 90, 150 and 210. The small equilateral triangle (below right) is used later on as a centre piece.

The kite and dart can be arranged in a certain way, to create an infinite tiling, radiating from a single point. Zig-zags are also possible.

Six fold rotational symmetry works using six ‘V’ tilings.

By combining groups of 60 and 90 degree angled sub tilings, larger patterns evolve. Bright yellow areas are basically just squares (a kite and dart for each).

‘V’ tilings can also overlap (white dart in centre below).

Other permutations used solely or in combination, are also possible. The fourth example below, is a mix from two out of the three patterns above it.

I tried creating a triangle but it left a hole. However, by filling this with a small equilateral triangle (in blue), a pattern can still emanate from it, without any triangle repetition.