We construct two Hirzebruch signature operators for some transitive invariantly oriented Lie algebroids A. Introducting a Riemannian structure in A we can define the *-Hodge operator, the codifferential and the first signature operator. Using the Hochschild-Serre spectral sequence for A and the Lusztig and Gromov examples we can define a second signature operator. Secondly, we describe a general algebraic approach to the *-Hodge operator, Hodge Theorem and Hirzebruch operator. We give applications to four fundamental examples concerning signature: classical example, Lie algebroids example, Lusztig example and Gromov example.