# Do Zero degree angles exist?

If an angle is the measure of distance between to points (Edit: Ok, admittedly bad phrasing. A measure of rotation between two intersecting lines, or points, etc.), is there such a thing as a zero degree angle? I asked a math nerd friend about this years ago and he still hates me for it but couldn't prove it either way at the time. Is this merely pedantic or delusional thoughts regarding definition of "angle" (Edit: based on answer below regarding Euclidian geometry, this all now seems very likely a colossal misunderstanding) or am I missing something? I do not think zero degree angles exist. Is there proof of this either way?

Edit: A better question would be why Euclidian zero degree angles were thought to not exist, or why he avoided them?

• Of course there is, it's just a flat angle, there's nothing mysterious or complicated here. Apr 13, 2016 at 21:29
• An angle is not really a measure of distance between two points. An angle is a measure or rotation of one ray with respect to the other ray (the rays share a common end point). When the 2 rays coincide and pointing towards the same direction, no rotation has taken place, and thus a zero degree angle is hereby introduced. Apr 13, 2016 at 21:29
• Do you think zero numbers exist? What's the difference?
– MPW
Apr 13, 2016 at 21:37
• Interestingly, discussion of what one would now call $0$ angles goes back to Hellenic times. They were concerned not about the angle between two straight lines, but between two curved "lines." Apr 13, 2016 at 21:38
• It's the angle that the hands of an analog clock make at 12:00.
– Dan
Oct 12 at 19:02

## 4 Answers

I suspect that behind this question lies the notion that an angle should look like some sort of wedge shape, created by two distinct lines meeting at a point, whereas a 'zero angle' just looks like part of a line.

In fact, that was very much the view taken by Euclid in Euclid's Elements, his seminal work of classical geometry, in which he states

"...a plane angle is the inclination of the lines to one another, when two lines in a plane meet one another, and are not lying in a straight-line."

This definition appears to rule out a zero angle, and indeed a 180-degree angle — in fact, the description of an angle by any number of degrees assumes a concept of angle measurement that Euclid avoided: instead he compared their size by mapping them onto each other, or added them by mapping them adjacently to one another, and was thus able to make remarkable progress.

The idea of measuring angles has been of practical interest for thousands of years for purposes such as astronomy, navigation and telling the time, but the use of numerical measurements as a tool for studying theoretical geometry didn't really take off until the work of Descartes and Fermat in the 17th century. Their work led to the study of geometry through algebraic functions and number, and the development of a more numerical understanding of angle was a natural part of that process.

In short, if you think of angles according to numerically-measured size, it is convenient to have a definition that extends along the number line, but if you're thinking of them from a purely spatial perspective, then such an extended definition is perhaps unnecessary and potentially confusing.

Zero degree angles do exist. They are the angle between a line and itself.

That's kind of like asking if a square is a rectangle. The answer depends on how you define angles and what you intend to do with them. Two examples:

In Euclidean geometry angle $\angle ABC$ consists of the union of the two non collinear rays $\overrightarrow{BA}$ and $\overrightarrow{BC}$. In this case, $0^\circ < m\angle ABC < 180^\circ$. The degenerate cases $0^\circ$ and $180^\circ$ are sometimes included to make some theorems easier to state.

In Trigonometry an angle, $\angle AOB$, consists of the union of an initial ray $\overrightarrow{OA}$, where $A$ is a point on the positive $x$-axis and a terminal ray, $\overrightarrow{OB}$. In this case $m\angle AOB$ can be any real number,

Wouldn't a zero angle be a point. Once you leave that point then you have created an acute, right, obtuse, straight, or reflex angle.

• No, If you leave a point you have created an angle? How? Mar 18, 2018 at 5:17