Why any heuristic which is an optimal solution to a relaxed problem is admissible and consistent?

Because heuristic does not necessarily need to provide the best optimal solution. If the solution is 'good enough' to handle the current situation without causing any fatal damages, a heuristic is considered as successful

And on top of that, the stake in a relaxed problem usually not high, so handling it more than a heuristic could be considered a waste ...