If asked to support euthanasia (I assume you mean voluntary euthanasia or assisted suicide, not a situation where some other person decides who lives and dies), I would make the following arguments:
- Personal freedom. A person should have the right to control their own life. This should include the right to decide when (perhaps within some limits) they would like to die.
- Prevention of unnecessary suffering. People who are suffering from terminal illnesses often have to deal with terrible pain for the last months of their lives. People who are in the early stages of Alzheimer's have to deal with the idea that they will become a total burden to their families without even being able to recognize or love those families. Both of these are terrible things and people should be able to avoid them.
- Economic reasons. Imagine if you had to choose between dying at a given time or living (at huge expense) for a few more months. Shouldn't you have the choice to leave your money for something (like your grandkids' college education) that means more to you than a few more months of life?
All of these are possible arguments for euthanasia.