He is good looking.
He is looking good.
How to understand "looking" in the first sentence? Is it exactly the same meaning as in #2?
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In the first sentence, the word looking is called a gerund. A gerund acts like a noun. The word good is an adjective in this sentence, and it is modifying the noun. The sentence means that he is a handsome man.
The second sentence use looking as a main verb in the verb phrase "is looking" and good is now an adverb modifying the verb phrase "is looking." The sentence means, informally, that he looks well or is doing well. You often hear it like this: Hey! You're looking good! I have won 500 dollars so far, and I'm looking good.
"Good-looking" is normally hyphenated, as it is an idiomatic expression. It is used to describe someone's natural looks, ie their facial features and perhaps their physique too. When someone is described as "good-looking" you would not expect that to change on a day-to-day basis (although good looks can eventually fade!) It is normally used to express physical attraction to another person.
To say someone is "looking good" is a comment on their current appearance. It usually takes in more than just their facial features or figure and includes their dress and grooming, and even their state of health. This is less permanent, and a person could look good one day and not the next. Also, this is less about expressing physical attraction and more about complimenting somebody - you might say that someone "looks good" after a period of illness, after some beneficial weight loss, or if they have made a special effort to groom for an occasion.