How is angular velocity of both these bodies equal here? Here there is John who is looking at a bus at which behind there is Harry. My teacher says that Angular velocity I.e(Theta / time)of the bus and Harry is equal when they appear to be at same point by Swaraj.
Now , my question is how is it possible since time and theta for both (John and Harry)appear to be different. Also , the arc Length of both are different Is it some Euclidean postulate my teacher used? I am not able to get which one is it here. My teacher did not give a reason for this.
I believe theta 1 and theta 2 are different as they appear to be. So will their time be different to reach at those points
 A: Consider these four pairs of arcs:

In each picture, the angles match: $\theta_{red} = \theta_{blue}$.
The arc length $L_{red}$ and $L_{blue}$ does vary, because $L = r \theta$, and the radiuses $r_{red}$ and $r_{blue}$ differ.
Angular velocity is defined as the rate of change in the angle $\theta$ with respect to time.  So, if we assume picture (1) with $\theta = 5°$ is taken at time $t = 1\,s$, and that picture (4) with $\theta = 20°$ is taken at time $t = 4\,s$, the average angular velocity is the same for objects represented by the blue arc as they are for objects represented by the red arcs:
$$\omega = \frac{\Delta \theta}{\Delta t} = \frac{20° ~-~ 5°}{4\,s ~ - ~ 1\,s} = \frac{15°}{3\,s} = 5°/s$$
The actual velocities (or speeds) do differ, average velocity being the distance travelled over the time taken.  Since the object represented by the blue arc travels a much longer distance than the one represented by the red arc in the same time, the velocity of the object represented by the blue arc is much higher than the velocity of the object represented by the red arc.  However, their angular velocities are the same.
A: The angular velocity can be given by $\frac{\Delta\theta}{\Delta t}$, i.e. rate of change of theta( the angle) with respect to time. Your confusion seems to arise from the fact that the theta's of the bus and K+G are different. I believe that the theta for both are the same and that is very clear from the diagram. Since both bus and K+G lie on the same line, they sweep the same angles. And since it stays at the same position w.r.t. the bus, it takes the same amount of time too.
Hence there angular velocity stays the same.
