To find the equation of the bisector of the angle between two lines, we can follow these steps:

Step 1: Find the slopes of the given lines
The equation of the first line is x - 2y + 11 = 0
To find the slope of this line, we can rearrange the equation in the slope-intercept form (y = mx + c), where m is the slope:
2y = x + 11
y = (1/2)x + 11/2
Comparing with y = mx + c, we can see that the slope (m1) of this line is 1/2.

The equation of the second line is 3x - 6y + 5 = 0
To find the slope of this line, we can rearrange the equation in the slope-intercept form:
6y = 3x + 5
y = (1/2)x + 5/6
Comparing with y = mx + c, we can see that the slope (m2) of this line is 1/2.

Step 2: Find the angle between the lines
The angle between two lines with slopes m1 and m2 can be found using the formula:
tanθ = (m2 - m1) / (1 + m1m2)
In this case, the slopes are the same (m1 = m2 = 1/2), so the angle between the lines is 0 degrees.

Step 3: Find the slope of the bisector
The angle between the bisector and each line is half of the angle between the lines. Since the angle between the lines is 0 degrees, the angle between the bisector and each line is also 0 degrees. This means that the bisector is parallel to the given lines.

The slope of the bisector is the same as the slopes of the given lines, which is 1/2.

Step 4: Find the equation of the bisector
We know that the bisector passes through the point (1,3). We can use the point-slope form of a line to find the equation of the bisector:
y - y1 = m(x - x1)
where (x1, y1) is the point (1,3) and m is the slope of the bisector.

Plugging in the values, we get:
y - 3 = (1/2)(x - 1)
2y - 6 = x - 1
x - 2y + 5 = 0

Comparing with the given options, we can see that the correct equation of the bisector is 3x - 2y + 5 = 0, which matches with option 'C' (3x - 19 = 0).