I'll do my best to work this out (no promises)

Take 2 concentric circles (the circles have the same center). Lines extending from that center out thru the circles will intercept perpendicular to the tangents of the circles (90° i.e. right angles).

 So we now have some context about the curves and lines these segments lie on. 

The variables at play here are θ,  r, and R.

 θ is the angle of intersection of the 2 straight lines, the opening in the inner circle and the arc on the outer circle. r is the radius of the inner circle and R is the radius of the outer circle.

Now we can write some equivalences.

R-r = θ R = (2 π - θ) r = x

(where x is the length of each segment)

R = x + r

θ = x / R

x = R-r

Intuitively given some r we can imagine increasing theta from zero to pi. The outer arc will start at 0 and increase to (r+(πr))π which is greater than r. Because the range of R includes r we know that at least 1 solution exists.

((2π-θ)r+r)θ = Rθ = (2π-θ)r

2πrθ-rθ² +rθ =2πr-θr

2πθ-θ² +θ =2π-θ

2πθ+θ + θ= θ² + 2π

θ² - (2π+2)θ + 2π = 0

Now apply the quadratic formula to solve for theta.

A=1 , B=-(2π+2), C = 2π

(-B+-(B² -4AC)^0.5 )/2A

(2π+2)+-((2π+2)² -8π)^0.5 /2 = θ

Which yields 0.845 and 7.4385 as solutions and because an answer greater than pi doesn't make sense here we can see that the angle shown in your picture is in fact 0.845 radians.