The problem of a slightly non-collinear, quasi-static crack growth is considered. Problems of this kind have been treated by a first-order perturbation method, in the contex of Mushkelishvili's complex potentials, by Banichuk, and Goldstein and Salganik. Recently, Cotterell and Rice have employed the same method and obtained a rather simple first-order expression for the stress intensity factors, which are then used to examine the crack growth path of a semi-infinite crack in an infinitely extended region. In the present paper we consider a solution method, which takes into account the effects of the geometry of the domain, i. e. finite outer boundaries, as well as the finite crack length. For this purpose we first calculate the first-order perturbation solution for a semiinfinite straight crack with a slightly branched and curved extension, which will be used as a fundamental solution of the problem. When a loading condition is given, the stress intensity factors at the extended crack tip can be deduced from its near tip field solution. Then we use an alternating solution scheme to establish the effect of the geometry of the domain, where the far field behavior of the fundamental solution plays an essential role. The first-order expressions for the stress intensity factors are obtained for a slightly branched and curved crack extension in a finite domain. The geometrical effect due to the crack extension, which appears in the expression of the stress intensity factors, is proportional to the length of the crack extension. An initial prediction of the crack growth path is obtained under a slightly non-symmetric loading condition. For this purpose we assume a locally symmetric deformation ahead of the crack tip; i. e. the Mode II stress intensity factor vanishes along the crack extension path. As a simple analytical example we consider the slightly slanted Griffith crack under the biaxial loading condition. The small inclination of the crack with respect to the principal stress directions, is considered as the imperfection (which causes a non-collinear crack growth), and the stability of the crack growth path is examined. Numerical applications of the present formulation to more complex problems will be made in subsequent works.