quantum mechanics - Does measuring destroy entanglement

Before measuring a quantum particle(photon) it exists in a superposition state, once we observe(measure) it, it settles in one of the possible states(destroying superposition). For entangled particles, does measuring destroy entanglement as well?

Answer

Yes: If you measure an entangled property, you destroy the entanglement, always.

I think @OndřejČernotík has already already nicely answered this one, but I think your question included the assumption that the measurement and the entanglement were talking about the same property, e.g. spin polarization. And if so, the answer is just a simple yes.

Superposition and entanglement slight variants of the same phenomenon, that being the ability of a quantum system to contain more than one possible state at the same time. In superposition the multiple states mostly share a single relatively small region in xyz space, such as an atom, whereas in entanglement the states may be quite large.

Here's as more specific example of that commonality, first given by Einstein: If you have a very large wave function and then find the particle at one distant part of that wavefunction, how is it that that probability of finding that same particle at some other large region of the wave function a light second "instantly" drops to zero? How did that distant part of the wave function "know" at an apparently superluminal velocity that the particle had already been found?

Einstein's example was just another form of entanglement, not of conserved angular momentum but of mass-energy. The universe insists on absolute conservation of both, so in both cases finding the property (angular momentum or mass-energy) at one well-defined location requires that that property be balanced out or removed from the rest of the universe, no matter how large the wave function has become.

In terms of superposition, such examples show that any quantum wave function contains entanglement, even if it is "just" a smooth, simple wave packet for the particle location. It's just that the entanglement issue does not show up clearly until you make the wave packet so large that its curious conflicts with the speed of light become apparent.