I need to find the obtuse angle between the following two lines

Question

I need to find the obtuse angle between two lines but I am not sure how to code it.

my two angles are 𝑥+√3𝑦=1 and (1−√3)𝑥+(1+√3)𝑦=8

I already did the calculations but im just not sure how to implement it, any help would be greatly appreciated.

Specifically what I need is to convert the two equations to slope intercept form, and then use that slope in the formula
Theta = arctan m2- m1/ 1+ m1*m2

Answer 1

Why slope intercept form? From the equations we immediately know their normal vectors are {1, Sqrt[3]} and {1 - Sqrt[3], 1 + Sqrt[3]}, so the angle between the lines is

VectorAngle[{1, Sqrt[3]}, {1 - Sqrt[3], 1 + Sqrt[3]}] // Simplify
(* π/4 *)

You need a obtuse angle, so we simply do π - π/4 and obtain 3 π/4. The whole process can be automated with:

angle…between…lines[eqns_, vars_] := 
 VectorAngle @@ CoefficientArrays[eqns, vars][[-1]] // Simplify

obtuse = If[# > Pi/2, #, Pi - #] &;
acute  = If[# < Pi/2, #, Pi - #] &;

Then we can calculate the angle directly from the equation:

obtuse@angle…between…lines[{x + √3 y == 1, (1 - √3) x + (1 + √3) y == 8}, {x, y}]
(* (3 π)/4 *)

If you need acute angle, use acute instead of obtuse in the code.

Answer 2

Or Differences the two points on the line to get the direction of line.

eqn1 = x + Sqrt[3] y == 1;
eqn2 = (1 - Sqrt[3]) x + (1 + Sqrt[3]) y == 8;
dir1 = {x, y} /. FindInstance[eqn1, {x, y}, Reals, 2] // 
   Differences;
dir2 = {x, y} /. FindInstance[eqn2, {x, y}, Reals, 2] // 
   Differences;
angle = VectorAngle[dir1[[1]], dir2[[1]]] // FullSimplify
Max[angle, π - angle]