Although all Christians believe abortion and euthanasia are both evils, different denominations react differently in the actual situations.
The Roman Catholic Church does not accept euthanasia or abortion whatever the circumstances. If someone in the Church were pregnant, they would do whatever they could to talk the woman out of having an abortion, and would suggest adoption as a possible option. However, although they are totally against abortion, they still offer pastoral care to women who have had abortions, which follows their belief of compassion. Although they are against euthanasia, they would say that the unnecessary prolonging of life is wrong.
The Methodist Church reacts quite differently to the situation. It also believes abortion is an evil but agrees that abortion should be legal. If there isn't enough money in the family, or the child has a terminal medical condition, the Methodist Church would accept abortion as the lesser of two evils.
Another of their arguments is that God lets 46% of eggs come to nothing, so why not this one? They don't accept euthanasia in any form because of the impracticalities that would be faced by doctors and nurses.
The Church of England also believes abortion and euthanasia are evils but it does accept abortion in some cases; if the girl can't cope or there isn't enough money. They also offer pastoral care for people who have had abortions. Although they don't accept euthanasia, they believe doctors should prolong life at all costs.
In the case of abortion, the Baptist Church would accept it if the girl couldn't cope or the child would be severely disabled, even though they also oppose abortion. They oppose euthanasia, but they think that when someone is brain-dead and most importantly, incapable of maintaining a relationship of any kind, Baptists agree it isn't wrong...