Extending a line, keeping the angle | But as part of a form!

http://helpx.adobe.com/illustrator/using/adjust-path-segments.html SAYS:
“Adjust the length or angle of straight segments
1. With the Direct Selection tool, select an anchor point on the segment you want to adjust.
2. Drag the anchor point to the desired position. Shift-drag to constrain the adjustment to multiples of 45°.”
Question: How do I proceed if I’d like to extend a line on one side by its endpoint, while I want to keep its original angle instead of 45° degrees?
And: assuming we’re not working with a standing-alone line, but as part of a construction, as you can see on the screenshot. The point should be move to the intersection on both lines (green and turquois).
I thought, maybe working with functions in GRAPHER – but I hope there is an easier way directly in ILLUSTRATOR.
Hints would be much appreciated.

Emil, this looks perfect, thank you a lot! I can not wait to give it a try !
May I ask a little detail while performing step nr. 3?:
When it comes to repeat the same drag at the lower part of the lozenge, the whole form is being addressed and scaled from the origine as soon as I start moving the point (it behaves as if the other points would also be selected).
I tryed the following combinations:
- click on the lowest point down in order to set the origine. Then click/hold on the second point (the one supposed to be dragged), and then drag the point.
- click on the lowest point down in order to set the origine. Then click/hold shift and then dragged the second point.
- click on the lowest point down in order to set the origine. Then just click/dragging the second point.
At all three tries, at least 3 sements of the lozenge were moved with the point, even once after deselecting the form. Do I omit any important detail?
Merci

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Extending a line, keeping the angle | But as part of a form!

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