Question 398287
Let x = # minutes after 6 pm when Anne came, and
y  = # minutes after 6 pm when Robert came.  
Then the sample space S = {(x,y): {{{0 < x}}}, {{{y < 60}}}} is the (open) square with vertices (0,0), (0,60), (60,0), and (60,60).
What is wanted is the intersection of the sample space S with the region satisfying
{{{abs(y-x) < 10}}}, or equivalently,

-10 < y-x < 10.

The boundaries of this double inequality are the lines y - x = -10 and y - x = 10.
All points between these two lines (excluding the lines themselves) satisfy the preceding double inequality.  Call this region R.

Then what is desired is the area of S intersection R. This area is equal to {{{A[S] - A[S-R]}}}, where {{{A[S]}}} = area of sample space S, and {{{A[S-R]}}} = the area of S-R.

Then the probability is equal to {{{(A[S] - A[S-R])/A[S]}}}.
Incidentally, S-R consists of two congruent triangles {{{T[1]}}} and {{{T[2]}}}.
{{{T[1]}}} has vertices (0,10), (50,0), (50,60)
{{{T[2]}}} has vertices (10,0), (60,0), (60,50)
(Each triangle has area {{{(1/2)*50^2}}}.)
The probability that Anne and Robert meet is then
{{{(60^2 - 2*(1/2)*50^2)/60^2 = (60^2 - 50^2)/60^2 = 11/36}}}