To find the acute angle bisector between two lines, we need to find the slopes of the lines and use the formula:

tan(θ) = |(m2 - m1) / (1 + m1 * m2)|

where m1 and m2 are the slopes of the lines.

The first line is given by 3x - 4y - 5 = 0. We can rearrange this equation to the slope-intercept form y = mx + b, where m is the slope:

4y = 3x - 5
y = (3/4)x - 5/4

So the slope of the first line is m1 = 3/4.

The second line is given by 5x + 12y - 26 = 0. Rearranging this equation, we get:

12y = -5x + 26
y = (-5/12)x + 13/6

So the slope of the second line is m2 = -5/12.

Now we can substitute these values into the formula for the tangent of the angle:

tan(θ) = |((-5/12) - (3/4)) / (1 + (3/4) * (-5/12))|
= |((-5/12) - (9/12)) / (1 - (15/48))|
= |((-14/12) / (33/48))|
= |(-7/6) / (11/16)|
= |(-7/6) * (16/11)|
= |(-112/66)|
= 56/33

Now, let's find the angle θ:

θ = atan(56/33)
θ ≈ 60.42 degrees

The acute angle bisector is the line that makes an angle of 60.42/2 = 30.21 degrees with the given lines.

To find the equation of the acute angle bisector line, we need to find its slope. Let's assume the slope of the acute angle bisector line is m.

Now we can use the formula for tan(θ) to find the slope:

tan(θ) = |(m - ((-5/12) + (3/4)) / (1 + ((-5/12) + (3/4)) * m)|
tan(30.21) = |(m - (9/48)) / (1 + (3/4) * m)|
0.576 = |(m - (9/48)) / (1 + (3/4) * m)|

Since the angle bisector is acute, the tangent is positive, so we can ignore the absolute value.

0.576 = (m - (9/48)) / (1 + (3/4) * m)

Simplifying the equation:

0.576 + 0.576 * (3/4) * m = m - (9/48)
0.576 + (0.432/4) * m = m - (9/48)
0.576 + 0.108m = m - 0.1875
0.108m - m = -0.1875 - 0.576
-0.892m = -0.7635
m ≈ 0.