Fourier transform of a tilted line function
The Fourier transform of polynomial functions exists in the domain of generalized functions, or distributions. The (non-unitary) Fourier transform of

$$f(t)=t^n\tag{1}$$

is given by (cf. entry 308 in this table)

$$F(\omega)=2\pi j^n\delta^{(n)}(\omega)\tag{2}$$

where $\delta^{(n)}$ is the $n^{th}$ generalized derivative of the Dirac delta impulse.

Consequently, the Fourier transform of

$$f(t)=at+b\tag{3}$$

is given by

$$F(\omega)=2\pi \left[ j\cdot a\, \delta'(\omega)+b\,\delta(\omega)\right]\tag{4}$$

If you want the Fourier transform of a line segment, i.e., of a line multiplied by a rectangular function, you need to convolve $(4)$ with an appropriately chosen sinc function.