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It depends on your dog's size, honestly. A golden eagle can attack small dogs (up to about 3-4 kg), but with medium to large dogs the risk is much lower because the odds aren't in the eagle's favor. My cousin has a German Shepherd and lives in the mountains where there are eagles, and he's never had any problems even though the dog is often outside - but he's never left it alone. If your dog is small, it's better not to let it roam alone in areas where you've seen the eagle, but if it's a decent-sized dog it's probably fine.
The main thing is to keep your dog closer to you when you're in areas where golden eagles are present. An eagle that spots a dog up close might see it as potential prey, especially if the dog is small or medium-sized. Golden eagles have powerful talons and impressive attack speed, so there really is a risk. If your dog is large (over 5-6 kilos I'd say, but it depends on the eagle) the chances of an actual attack drop quite a bit, but it's still a situation you should avoid.
The ideal thing is to never leave your dog alone outdoors in mountainous or hilly areas where golden eagles nest. If you go hiking in those places, keep it on a leash as much as possible and stay alert. If you notice eagles circling overhead or in your area, immediately pull the dog toward an area with thick vegetation or some shelter. In most cases the birds avoid direct confrontation if they see the dog is with a person, because they know it would become dangerous for them.
If there's already been a close encounter but your dog is fine with no injuries, you can relax. The eagle probably assessed the risk and took off. Still check that it doesn't have scratches or bites, and if you notice strange behavior in the following days (excessive fear, refusing to go outside) then it might have had a serious fright and maybe deserves a vet visit.
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