You have to first combine the 2 objects with the proboolean compound object to make edges or vertices that you can snap to. They are now separate objects, so max doesn't recognize any separate edges/vertices where they intersect because there are none. So select one of the objects and then select Create Tab> Compound Objects> Proboolean. Then select "union" as the boolean operation and the "Start Picking" and then select the other object. After that you can put an editpoly modifier on it and make any changes you want to it and you will have edges where the union operation joined the 2 objects. Look up Proboolean in the max docs for a more detailed account of the different operations. Hope it helps.
@10DSpace solution is correct for 3D objects. To "snap" to the intersection of the two lines in your drawing you must also first attach one to the other via Modify / Line / SPline sub-object. At this stage you can use the trim option within the spline sub-object options. This will yield a line with an endpoint at the intersection. YOu can then use this endpoint as the intersection. Awkward and messy? Absolutely! Max is not designed to easily create precise models (the database is limited precision). Rather than precisely snapping to an intersection point the intended workflow is to eyeball the intersection with perhaps a snap setting of Edge/Segment to ensure the the intersection point is on at least one of the two lines.
