Question

Question #567c3

Answer 1

There are two different directions of the lines equally inclined to both axes. For each direction there are infinite number of lines parallel to each other.

Explanation:

Consider a right triangle formed by two axes and any line equally inclined to both axes. Since this line (a hypotenuse) should be equally inclined to both axes, a triangle must have two congruent acute angles of `45^o` each.
An example of this line is the one going fro point `(0,1)` to `(1,0)`, which we call base line.

Any other line parallel to the base one we considered above will also be equally inclined to both axes.

Since we can construct `4` different right isosceles triangles in `4` quadrants, we have `4` base lines equally inclined to both axes:
line `a` from `(0,1)` to `(1,0)`
line `b` from `(0,1)` to `(-1,0)`
line `c` from `(0,-1)` to `(1,0)`
line `d` from `(0,-1)` to `(-1,0)`

But opposite triangles (1st and 3rd quadrant as well as 2nd and 4th) have these lines parallel to each other:
`a` `||` `c` and `b` `||` `d`.
Therefore, there are only two different directions of our lines that form congruent angles with axes. For each such direction there are infinite number of lines parallel to the base line and forming congruent angles with axes.