The first thing is to get the attention of the animal and for that you can softy talk to it or make a soft sound with your mouth. However, be very cautious not to frighten the animal.
Once you have the attention of the cat or dog, walk towards it taking small steps. Do not under any circumstances pace yourself towards it, as the animal will consider this as a threat and will run off.
After getting close to the cat or dog, stop and wait for the animal to make the final approach towards you. It is very important that the animal feels comfortable and safe enough to walk towards you. During this time do not make an extended eye contact as dogs and cats consider this threatening. Keep looking at the animal but without making an eye contact.
You can also through food near the animal, in order to initiate its approach towards you. However, do not throw the food at it, as you will just scare it off.
Reach out your hand towards the animal and let it sniff it, avoid directing your fingers as they might get bitten. Instead, hold the hand in such a way that the animal’s mouth can only reach the palm.
Be patient and allow the cat or dog to complete its examination before you try to pet it. Once the animal is done and chooses to stay, you can then move your hand towards the shoulder very gently. Do not pet the animal quickly or on the head as it might react by biting you or clawing you.
The more you pet, cats and dogs will get more comfortable and will lower their guard down, which will be a sign of acceptance towards you.
You can now reach for the ID tag to see who the animal belongs to.