The proof is simple in one direction . . .
An angle θ such that 0° < θ < 90°
can be constructed with a ruler if and only if tan θ is a rational number.
If is a rational number , where are positive integers,
. . we can construct angle
Code:* R * | * | * | a * | * θ | - - * - - - - - - - - * - - P b Q
On a horizontal line, mark points P and Q so that: PQ = b
At Q, erect a perpendicular QR so that: QR = a
Draw line segment PR.
. . Then: ./ RPQ = θ