Is $540^\circ$ a straight angle? The usual definition of a straight angle is a $180^\circ$ angle. however, because a $540^\circ$ angle is also the same shape, is it a straight angle as well? 
 A: $540^\circ$ is coterminal with $180^\circ$, so I would just say it's "coterminal with a straight angle". But angles with more than $360^\circ$ are not called straight. 
A: If I follow Euclid's five postulates and the five common notions
Postulate:
1. A straight line segment can be drawn joining any two points.
Common notion:
4. Things which coincide with one another equal one another
Then I would say yes. But I have no idea if this mode of thinking is incorrect.
A: The angle is the same as 180, unless you are using some device that counts turns.  Such are known in the more esoteric parts of geometry, but for ordinary geometry, you just done a 360 spin and facing the right direction.
A: It should depend on your definition of straight angle. The standard definition of straight angle, insofar as I'm aware, is one of measure $180^\circ$. In that sense, no, $540^\circ$ is not a straight angle simply due to being a strictly different rotation than $180^\circ$.
However, as others have pointed out, $180^\circ$ and $540^\circ$ are coterminal. That is, they share the same initial and terminal sides. If your definition of a straight angle is that the initial and terminal sides form a straight line, then $540^\circ$ would surely satisfy this. I think this latter definition is more intuitive as to what is meant by "straight angle", but I also believe the former is more standard convention. 
I hope this helps. 
A: Really, 540˚ is not an angle, but a measure of rotation, as an angle is the measurement of the final product, after the rotation has been applied. A rotation of 540˚ will give an angle of 180˚, but is not the same thing. If you were to rotate a dial by 540˚, It would give you a different result than if you only rotated it 180˚ - a straight angle.
A: Ultimately this is (as Thomas Andrews comments) a question of definitions, like asking "Is $1$ prime?" The "right answer" depends on what definition (of "numerical value of angle") best conforms to the desired usage, or is most convenient for stating and proving theorems. Here are sample arguments in favor of each answer:


*

*According to Wikipedia, an angle in planar geometry is a geometric figure formed by two rays. ("Planar geometry" has no precise definition on that page, but the examples given clearly suggest an "approximately/infinitesimally Euclidean" setting, such as spherical or hyperbolic geometry, in which "one full turn" has the same meaning at each point.) In this sense, an angle is not a real number, but a real number modulo one full turn. Thus "$540$ degrees" is a "straight angle", because a straight angle is a pair of rays pointing in opposite directions.

*On the other hand, one has theorems such as "the total interior angle of a Euclidean triangle is equal to a straight angle", in which angles are implicitly given numerical measures. Here, an angle of $540$ degrees is certainly not a "straight angle". (One might attempt to nitpick: In the diagram, how do we know angle $\alpha$ (say) isn't one full turn plus the angle shown? This objection would be perverse: In hyperbolic or spherical geometry, the angular defect of a geodesic triangle is proportional to the triangle's area. You can't play word games with angle measurement in those settings without losing valuable theorems.)

To coin (non-standard) terms, define:


*

*A geometric angle at $p$ to be a figure formed by two rays originating at $p$. The measure of a geometric angle is an element of the circle $\mathbf{R}/(\text{one full turn})\mathbf{Z}$, e.g., $\mathbf{R}/(2\pi \mathbf{Z})$ if angles are measured in radians.

*An algebraic angle at $p$ to be a continuous path in the space of rays originating at $p$. The measure of an algebraic angle is a real number, the "total winding" of the path, i.e., "how far (including "orientation", or "direction of rotation") the initial ray turns, viewing one turn as distinct from doing nothing".
There is an obvious mapping from algebraic angles to geometric angles (sending a path of rays to the geometric angle subtended by the initial and final rays), but no "continuous section", i.e., no way of continuously assigning a real number as measurement of a geometric angle.
In this terminology, again, $540$ degrees is straight as a geometric angle, but not straight as an algebraic angle. (This much seems non-controversial, and at least implicitly present in both the question and all prior answers.)
The notion of algebraic angle strikes me as the more fruitful of the two, and not merely because it contains strictly more information. For instance, the physical properties of an electromagnetic coil depend on the number of turns: The laws of electromagnetism arguably also favor algebraic angles. Identification of additional examples is left as a pleasant exercise.
So, if you ask me, "No, an angle of $540$ degrees is not a straight angle."
