Angle between the two straight lines:
To find the angle between two straight lines, we need to determine the slopes of the lines. The slopes of the lines can be found by rearranging the given equations into the slope-intercept form (y = mx + c), where m represents the slope.

Given equations:
1) 3x = 4y + 7
2) 5y = 12x + 6

Let's rearrange equation 1 to slope-intercept form:
3x - 4y = 7
-4y = -3x + 7
y = (3/4)x - 7/4

And equation 2:
5y = 12x + 6
y = (12/5)x + 6/5

Now, we can compare the slopes of the two lines. The slopes of the lines are equal to the coefficients of x in the slope-intercept form.

The slope of the first line (m1) = 3/4
The slope of the second line (m2) = 12/5

Using the formula:
The formula to find the angle (θ) between two lines with slopes m1 and m2 is given by:
tan(θ) = |(m1 - m2) / (1 + m1 * m2)|

Let's substitute the values:
tan(θ) = |(3/4 - 12/5) / (1 + (3/4)(12/5))|
= |(-33/20) / (1 + 9/10)|
= |-33/20 / 19/10|
= |-33/20 * 10/19|
= |-33/19|

To find the angle (θ), we can take the inverse tangent (arctan) of |-33/19|:
θ = arctan(33/19)
θ ≈ 59.82°

Equations passing through (4,5) and making equal angles:
To find the equations of the lines passing through the point (4,5) and making equal angles with the given lines, we need to use the concept of perpendicular lines.

Perpendicular lines:
Perpendicular lines have negative reciprocal slopes. The negative reciprocal of a slope m is -1/m.

The negative reciprocal of the first line's slope (3/4) is -4/3.
The negative reciprocal of the second line's slope (12/5) is -5/12.

Equations of lines:
The equations of the lines passing through (4,5) and having slopes -4/3 and -5/12 can be found using the slope-intercept form.

For the line with slope -4/3:
y - 5 = (-4/3)(x - 4)
y - 5 = (-4/3)x + 16/3
y = (-4/3)x + 16/3 + 15/3
y = (-4/3)x + 31/3

For the line with slope -5/12:
y - 5 = (-5/12)(x - 4)
y - 5 = (-5/12)x + 20/12
y = (-5/12)x +