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The definition of an angle in Euclidean geometry is given as follows:

Angle. Definition: A shape, formed by two lines or rays diverging from a common point (the vertex).

However, the definition nowhere states how to define the measurement for an angle. With a bit more digging/ thinking its obvious that the definition for a measurement is something like:

Draw a unit circle around the vertex. The angle is then measured as the arc length between the intersection of the two lines and the circle. (in radians)

That is how measuring an angle seems to be always defined, but given that the definition of an angle does't outright state that it is the only way of measurement, I wonder if other measurements could be used that are consistent with Euclidean geometry. For example:

enter image description here (So the measurement of angle $\alpha$ in each case is the length of the hand drawn arrow - as if it was perfectly drawn on top of the arc/lines).

Now, for any non-circle based measurement, it would seem that whenever the "measurement apparatus" is spun the angles change, which means that they have to be always used parallel to some reference line. On the other hand, calculating sin() etc. becomes really easy.

Question: Is there something seriously wrong in altering the methodology of how angle is measured, that might break Euclidean geometry? Is the "circle" definition of an angle the only valid one?

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    Euclidean geometry is isotropic, which is a big asset. – 2017-01-08

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The addition of two angles is defined without reference to measure. The definition of measure $m(\alpha)$ of an angle $\alpha$, to be meaningful, should satisfy the relation $m(\alpha+\beta)=m(\alpha)+m(\beta)$, which your alternate definitions don't.

EDIT.

As the above does not seem to be clear enough, let me give an example (see diagram below). Take a point $C$ on the $x$ axis and a ray $OD$ with $OD=OC$. Let $E$ be the midpoint of $CD$: it is a well-known theorem in elementary geometry that in isosceles triangle $OCD$ the median $OE$ also bisects angle $\angle COD$, so that $\angle COD=2\angle COE$.

But with your triangular definition, for instance, the measure of $\angle COD$ is not, in general, twice the measure of $\angle COE$, because $AE'>E'D'$.

enter image description here

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    I don't see how $m(\alpha+\beta)=m(\alpha)+m(\beta)$ wouldn't hold for the cases above (as long as a reference was defined, if not even $m(\alpha)=m(\alpha)$ would not hold). In any case, is there any reference that proves the important properties of the circle based method? The materials I have skip right over it... – 2017-01-08
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    In the examples you provide, for instance, the measure of an angle of 60° is not the double of the measure of an angle of 30°. – 2017-01-08
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    To be honest, I am not following. How can the measure even be a function of $\alpha$ if that's the very thing we are trying to measure/define? Do you have any references? – 2017-01-08
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    If ray $OC$ is inside angle $\angle AOB$, then by definition $\angle AOB=\angle AOC+\angle COB$. A sound definition of measure should then satisfy $m(\angle AOB)=m(\angle AOC)+m(\angle COB)$. A simple sketch should convince you that this does not hold for the examples you give. – 2017-01-08
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    See also my edited answer. – 2017-01-08