if figure (b) is the output of signal in figure (a) in an LTI system, how can I get the output of another signal (one in figure c) in an LTI system?
The system is linear and time invariant. So $\alpha x_1+ \beta x_2\to \alpha y_1+ \beta y_2 $, and also a time-shift in the input will lead to the same time-shift of the output. In this case $x_1$ is the rectangle and $y_1$ is the triangle. The input in the question is $(x_1(t)-x_1(t-2)$, so the output will be $\left (y_1(t)-y_1(t-2)\right )$.