How the triangle definition relates with its existence In Plane and solid geometry by Fletcher Durell(p.32), it states that:

A triangle is a portion of plane bounded by three straight lines, as
  the triangle ABC".

How does this statement relates with the existence of a triangle. For example, how could three sides be chosen so the figure obtained is a portion of a plane bounded by three straight lines?
 A: In order for a portion of the plane to be bounded by three straight lines, the lines in question need only:


*

*not be parallel to each other

*intersect in more than one point (i.e. 3 points; if you wish the area of the bounded region to be nonzero)*


*Proofs available upon request.
Examples
As a construction:
How to make a triangle (in a plane):
Choose (or draw) 2 non-parallel lines. These lines intersect in a point.
Choose an additional line that:


*

*Does not intersect the aforementioned point. (unless you want the bounded area to be zero)

*Is not parallel to either of the prior 2 lines.


For example, in the Cartesian plane:
If you use the x and y axis for the first two lines, then they intersect in the origin. If the third line you choose isn't parallel to either axis, and doesn't run through the origin, it can be written in the form y=ax+b or x=cy+d, such that  a, b, c, and d are $\neq 0$.
For another construction:
Choose two non-parallel lines.
Pick a point on each line, other than the intersection, and draw a line through those points.
The area bounded by the lines ought to be between the third line you draw, and the intersection (of the first two lines you chose).
There are some illustrated examples here: 
https://en.wikipedia.org/wiki/Parallel_postulate and here:
http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf under Proposition 1.
I understand this book (Euclid's Elements) is filled with constructions and constructive proofs.
A Pictures of 3 lines, no 2 of which are parallel:

Note that if all three lines intersected in the same place, there wouldn't be any bounded region of non-zero area.
