I'm teaching mathematics on my free time for young pupils. Once I have seen that they denote angles like $\angle ABC$. But sometimes I have difficulties to understand whether they mean an angle or its explementary angle, especially in problems containing angles in a circle. Should I worry about this or should I teach them some other notation, like $\angle ABC$ is the explementary angle of $\angle CBA$ or the $\angle ABC$ where the some given point $D$ is between the rays $BA$ and $BC$? And also if some most talented pupils wants to participate in mathematics competitions, should he or she write the answers so that one knows exactly which angle he or she means or is it better to save a few seconds just by leaving the reader to think which one angle he or she means?

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    $\begingroup$ Interpreting $\angle ABC$ as the counterclockwise angle that maps $BC$ to $AC$ is an important convention. With that (or the opposite) convention, the ambiguity is removed. $\endgroup$ Commented Dec 11, 2016 at 22:23
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    $\begingroup$ @JosephO'Rourke, in the elementary school where I taught, the books teach that $\angle ABC$, is equivalent to $\angle CBA$. Either way the angle always stands for the angle that is 180 degrees or less. Not sure where this convention came from, but it is clearly different than your convention. $\endgroup$
    – Amy B
    Commented Dec 12, 2016 at 14:23
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    $\begingroup$ @AmyB: Maybe my convention comes later in the educational pipeline, fitting in with the right-hand-rule for e.g. cross-products. $\endgroup$ Commented Dec 12, 2016 at 14:58
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    $\begingroup$ @JosephO'Rourke Considering that the OP is teaching "young pupils", the ordinary convention for the Euclidean program (ie, the protractor postulate) of measuring angles from 0 to 180 degrees is appropriate here. I have never encountered a counterclockwise convention in any school geometry textbook or competition. Also, what do you mean that an angle "maps" $BC$ to $AC$? $\endgroup$
    – Andrew
    Commented Dec 14, 2016 at 18:31
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    $\begingroup$ @Andrew: Oops, typo. Maps $BC$ to $BA$ = rotates $BC$ about $B$ counterclockwise, until it lies along $BA$. $\endgroup$ Commented Dec 14, 2016 at 20:29

1 Answer 1


The answer is:

  1. You should not worry about it.
  2. Yes, you should let the reader interpret the angle according to the context.

There is no defined convention for notation of explementary angles. Because it's not of much practical importance. However if you skim through some papers, you'll notice that generally, angles are taken to be the ones less than 180 i.e., unless explementary is explicitly mentioned. In other words explementary angles are often taken to be greater than 180.

Also, when it's imperative to explicate which angle is being referred to, it's better to label the corresponding arc, and mention the angle as the one spanning through the specified arc.

  • $\begingroup$ Added the last part because putting a point inside the region is too cumbersome and not appropriate. Labelling arc, on the other hand is something that we do anyway. $\endgroup$
    – yathish
    Commented Aug 27, 2017 at 3:38
  • $\begingroup$ Describe the angle as a deformation of a parallelogram. Ttell the pupils that the angle is always inside and the explementary angle is always outside of the parallelogram. You get rid of disambiguities, no need for degrees, or radians. You may have to draw a different parallelogram, but the angle will be always well defined. A written notation may be naturally tied to the concept anytime later. $\endgroup$
    – Thinkeye
    Commented Aug 28, 2017 at 14:55

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