I have noticed that a cat and dog share the same radical, which is 犭dog (quǎn).
So, if we break the words down:
猫 cat (māo) = 犭dog (quǎn) + 艹 cǎo (grass) + 田 tián (field)
狗 dog (gǒu) = 犭dog (quǎn) + 勹 bāo (wrap) + 口 kǒu (mouth)
I have two questions:
1) Why a dog radical has the dog in it with other components?
2) Why a cat has the dog radical it in?
These words are not difficult to remember. For a cat, I use 'meow', which is somehow similar to māo. For a dog, I stick to 'Go, go, dog!', which sounds a bit like gǒu.
However, I am curious about the origin of these words.