the properties of contour integrals
Contour integrals obey the same comfortable bookkeeping rules you already trust from ordinary integration, because each contour integral is, underneath, an ordinary integral in the parameter. Knowing these rules lets you break a hard integral into easy pieces and recombine them, which is most of the day-to-day work of the subject.
Three rules carry the load. Linearity: the integral of a f + b g equals a times the integral of f plus b times the integral of g, for constants a and b. Reversal: if you reverse the orientation of the path, the integral changes sign — integrating along C from start to end and along the reverse of C cancel out. Additivity over concatenation: if a contour C is C1 followed by C2 (the end of C1 is the start of C2), then the integral over C is the integral over C1 plus the integral over C2. So you may split a path at any convenient point and add up the pieces.
These properties are exactly what let you compute over a polygon edge by edge, or argue that two integrals are equal by joining one to the reverse of the other to form a closed loop. They hold for any continuous integrand — no holomorphy required — because they are inherited directly from the linearity, sign, and splitting rules of the ordinary integral in t. The deeper facts (path-independence, vanishing loops) need holomorphy; these basic ones do not.
To integrate f over a triangle with vertices p, q, r counterclockwise, write the boundary as three segments [p to q], [q to r], [r to p]. By additivity the integral over the triangle is the sum of the three segment integrals; by reversal, swapping a segment's direction would negate its contribution.
Additivity over concatenation lets a polygon integral be assembled edge by edge.
A handy corollary of reversal: the integral of f over a path joined to the reverse of another path is the difference of the two integrals — the standard trick for comparing two routes between the same endpoints.