We will assume that at 12 noon the angle of the sun is 90°

Let x = the angle of the sun in degrees
Let y = the time of day in hours after midnight

Then when x is 15, y is 8, and when x = 90, y is 12

So we have two points on a line, (15,8) and (90,12)

So we find the equation of the line that passes through
those two points:

Slope formula:
m = 
where (x1,y1) = (15,8)
and where (x2,y2) = (90,12)

m = 
m = 
m = 

Point-slope formula:

y - y1 = m(x - x1)
where (x1,y1) = (15,8)

y - 8 = (x - 15)

y - 8 = x - 

    y = x -  + 8

    y = x -  + 

    y = x + 

So if we substitute the angle of the sun in that
equation, we will get the time of day.

So we substitute x=55

    y = (55) + 
 
    y =  =  =  =  =  

ths of an hour is 8 minutes,

so the time is 10:08AM


-------------------------
FYI, after 12 noon the sun starts going down, so to use the formula
in the afternoon, you must subtract the angle from 180 degrees, Then
the formula will give you military time, which you will convert to
ordinary time by subtracting 12 hours.  

For instance, if the angle of the sun were 55° and it were in the 
afternoon instead of the morning, we would subtract 180°-55° and 
get 125, we would substitute that in the equation and get  or
 and we would subtract 12 hours and get  
so the time would be 1:52PM, which is the same amount of time AFTER
12 noon that 10:08AM (the answer to your problem) was BEFORE 12 noon.  

Edwin