The equation x+y=6 can be written in slope-intercept form,
y = -1x +6
Which is a line with a slope of -1 (i.e. top left to bottom right) and intercepts the y axis at 6.
In order to determine whether a this is parallel to another line, you need to compare slopes - if the slopes are exactly equal, then the lines are paralell.
To determine if 2 lines are perpendicular, the product of the slopes is -1, so y = -1x + 6 will be perpendicular to any line which has a slope of +1 (since -1 x 1 = -1).
You haven't given us the equations of the other lines, but you can use the above criteria to determine whether other lines are parallel or perpendicular.
You can visualize these lines here:
http://statsfiddle.info/Graph/SolveLinearEquation?LinearSolutionType=SlopeAndYIntercept&Slope=-1&YIntercept=6