Yes, you can. Just put the vertexcoords of every vertex on the triangle (better use a quad, with 2 triangles) in the position of the texture.
Example:
The points are the vertex. So put their uvw in the position of the texture you want.
Example:
.-----. | / | | / | ./----.
The points are the vertex. So put their uvw in the position of the texture you want.