My mom argues with other dog owners, I find it really embarrassing?

Me and my mom are looking after my brothers dogs for 2 weeks. One of them is quite reactive and barks a lot at other dogs when they come near him. Today, we were walking with them. They had their leads on and we were approached by an off lead dog and it’s owner. Our reactive dog was on the lead and it was barking and barking as the other dog approached and my mom asked the owner is she could put her dog on the lead. She didn’t and said it would make things worse if she put her dog on the lead and that her dog was friendly. It turned into a bit of an argument with me in the middle. I felt pretty embarrassed to be honest and a little bit shaken. I’ve been with my mom before and this has happened several times. I just wanted to vent. My brother will be back for them soon, but what is the best way to handle this type of situation?

My mom argues with other dog owners, I find it really embarrassing?
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