Proof relating to Measurement of angle The question asks to prove that radian is a constant angle.

First, I didn't understand the question and what is way to prove that the radian is a constant angle.
Your question is not so clear. 

Anyway you can proceed from figure similarity comparisons bringing in other figures that are implied here.

Draw a series of circular arcs enclosed between two radial lines making an angle $\theta $ between the lines or arms.  Join the chords. When triangles or sectors are placed so that circle centers are same, it will be noticed that the ratio of corresponding sides or arcs is same. That is,

$$ \frac{Arc_1}{Arc_2} = \frac{Arm_1}{Arm_2}$$

This ratio is called proportion in its simplest form. It is a constant.It is saying the same thing as:

$$ \frac{Arc_1}{Arm_1} = \frac{Arc_2}{Arm_2}$$

This ratio is called angle $\theta$ here,  which must be also... and likewise be constant.
Let $O$ be the centre of a circle and radius $OR = r$. If we take an arc $AB = OA = r$, then by definition, $∠AOB =1$ radian.
Let $AO$ be produced to meet the circle at the point $C$. Then the length of the arc $ABC$ half the circumference and $∠AOC$, the angle at the centre subtended by this arc $= a$ straight angle $=$ two right angles.
Now if we take the ratio of the two arcs and that of the two angles, we have
arc $AB/$arc $ABC = r/(1/2 × 2\cdot\pi\cdot r) = 1/ \pi$
$∠AOB/∠AOC = 1$ radian$/2$ right angles
But in geometry, we can show that an arc of a circle is proportional to the angle it subtends at the centre of the circle.
Therefore, $∠AOB/∠AOC =$ arc $AB/$arc $ABC$
or, $1$ radian$/2$ right angles $= 1/\pi$
Therefore, $1$ radian $= 2/\pi$ right angles
This is constant as both $2$ right angles and $\pi$ are constants.
The approximate value of $\pi$ is taken as $22/7$ for calculation