Participles in Classical Greek are widely used, are in the same case/number/gender as the element to which they are referred and may have two different functions: nominal or verbal one.
In the first case, the participle is always preceded by an article and it may be substantivized, acting as a noun, or attributive, acting as an adjective and related to a noun and usually translated with a relative clause.
In the verbal function, the participle is not preceded by an article and it may be verbal, acting as a subordinate clause (causal, temporal, concessive, conditional, purpose); in alternative it may be predicative, which is used after verbs such as to see, to hear, to announce, to declare and helps explaining what you are seeing/hearing/declaring.