What is an Angle?

[Anonymous]

The Measurement and Data progression defines it as the union of two rays at a common endpoint. If this is the case, then a pair of rays AB and AC that both originate at point A would constitute a unique angle that has multiple measurements (x, 360 – x, and others if not confined to a full circle). Verbiage from other sources (e.g., Wikipedia) refers to angles as being enclosed by the rays and therefore not the rays themselves. Under this notion, the pair of rays forms two angles: one measuring x and the other measuring 360 – x.

While this might seem to be a clarification of little consequence, being sure of this would be helpful when writing problems for Common Core. For example, suppose we want students to identify the measure of an angle as 60 degrees. For emphasis, we would have the 60-degree space associated with this angle marked with an arc or some sort of shading and direct students to the “marked angle” in the text. But if the angle is merely the union of two rays, then “300 degrees” would technically be an accurate measurement of the marked angle in which case we should not include it as an answer choice (because 300 is one of the measures of the angle regardless of any interior markings).

If, instead, the progression took a less definitive approach explaining that angles are “formed by these rays” and not the combination of rays themselves (similar to what I see in the wording of standard 4.MD.5), writers such as myself could more freely refer to the rotational space on one side of the pair of rays as the angle in question without any technical violation. Moreover, including the aforementioned “300 degrees” as an incorrect answer choice would not present the same issue, since it would not describe the angle, a.k.a. rotational space, that is marked. Under the present definition in the progression, 300 degrees DOES describe/measure the angle, a.k.a. pair of rays, making it less appropriate to entice students with it.

Thank you for your feedback and insight.

[Bill McCallum]

I think you mean the Geometric Measurement Progression, right? The full quote is “An angle is the union of two rays, a and b, with the same initial point
P. The rays can be made to coincide by rotating one to the other about P; this rotation determines the size of the angle between a and b.” So you need to specify a direction from one ray to the other in addition to the rays. I think the meaning is clear enough, but a more formal definition would be something like “An angle is defined to be the union of two rays, a and b, with the same initial point P, along with a direction of rotation from one ray to the other.” Would that be better? I worry that it would sacrifice clarity for precision.

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