How can we tell the cross point of two half-open lines given by starting points and angles? Consider having two lines given by their start points ($A$ and $B$) and their angle with respect to the $x$-axis.

These are half-open lines, and the intersection may occur only after the starting points. For example, they do not intersect in the given image if $a>b$ or $a$ is negative.

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*How can we tell if they cross from their angles?

*What is the fastest way to find the intersection?

I have two options: First, using an imaginary far endpoint, calculate the intersection of line segments. Second, treat them as two vectors.
But I think both are overkill, as a simpler solution might be available using the angles (instead of the endpoints).

Let Equation of line from A is:
$y=a_1 x+b_1$
And that of B is:
$y=a_2 x +b_2$
Where $a_1=tan (a)$ and $a_2=tan (b)$ . These lines intersect if the angle between them is less than $180^o$, that means $a\neq b$ or they are not parallel. This angle can be found from following formula:
$$tan(\theta)=\frac {a_2-a_1}{1+a_1a_2}$$
For second question you have to solve following system of equation which gives the ordinates of point of intersection:
$\begin{cases}y=a_1x+b_1 \\y=a_2x+b_2\end{cases}$
