Definition 11

An obtuse angle is an angle greater than a right angle.

Definition 12

An acute angle is an angle less than a right angle.


The angle ABD in the figure is an obtuse angle. It is greater than a right angle, but less than two right angles. Recall that Euclid required that any angle be less than two right angles.

The angle CBD is an acute angle. It is less than a right angle.

Note that there is no requirement that the angle be rectilinear, indeed, the horn angles mentioned before are not rectilinear, but they are less than right angles, and so are acute (notwithstanding Proclus’ remarks to the contrary).

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With these definitions, we see another aspect of magnitudes, namely, two magnitudes of the same kind, such as two angles, can be compared for size. Euclid frequently uses what is known as the law of trichotomy: given two magnitudes F and G of the same kind, exactly one of the following three situations hold, F is less than G, F equals G, or F is greater than G. See the comments after the Common Notions for more discussion on magnitudes and the law of trichotomy.