"Do you know why they are angry?"
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You don't use 是 to describe a noun with an adjective (The cat is white. / His house is big.). You only use it to define what something is (usually it's "subject + 是 + noun", for example: I am a student. / He is my brother).
For describing a noun with an adjective you either use 很, 非常, 太。。。了, (and more), or nothing at all, like in this case. 很 means very, but I think Chinese speakers simply use it to connect nouns with adjectives because it may sound off not to do it? The latter is just my own speculation.
The less intuitive answer is that adjectives act as verbs. Like 生气 would mean "to be angry".
吗 indicates that this is a yes/no question, so it's extremely important. Note the difference between the first and second sentences:
你喜欢去公园。--> You like to go to the park.
你喜欢去公园吗？ --> Do you like to go to the park?
你喜不喜欢去公园? --> Do you like to go to the park?
Likewise, the meaning of this sentence changes dramatically without 吗.
你知道他们为什么生气吗？--> Do you know why they're angry?
你知道他们为什么生气。--> You know why they're angry.
The Collins Dictionary reference for 吗 can be found here: https://www.collinsdictionary.com/dictionary/chinese-english/%E5%90%97
You have to include 吗 because this is a yes/no question. The question here isn't "Why...?". The question is "Do you know ...?".
It may be a bit confusing because the "Wh...?" questions in English always start with the WH word, whereas in Chinese the WH words (为什么，什么， 什么时候，谁， 哪儿， ...) aren't always the first word in the sentence, even when it's a WH question.