Dependent Type Semantics (DTS; Bekki and Mineshima, 2017) is a proof-theoretic compositional dynamic semantics based on Dependent Type Theory. The semantic representations for declarative sentences in DTS are types, based on the propositions-as-types paradigm. While type-theoretic semantics for natural language based on dependent type theory has been developed by many authors, how to assign semantic representations to interrogative sentences has been a non-trivial problem. In this study, we show how to provide the semantics of interrogative sentences in DTS. The basic idea is to assign the same type to both declarative sentences and interrogative sentences, partly building on the recent proposal in Inquisitive Semantics. We use Combinatory Categorial Grammar (CCG) as a syntactic component of DTS and implement our compositional semantics for interrogative sentences using ccg2lambda, a semantic parsing platform based on CCG. Based on the idea that the relationship between questions and answers can be formulated as the task of Recognizing Textual Entailment (RTE), we implement our inference system using proof assistant Coq and show that our system can deal with a wide range of question-answer relationships discussed in the formal semantics literature, including those with polar questions, alternative questions, and wh-questions.