Final answer:
Mr. Puncto likely encountered an error, as the sum of angles in a Euclidean triangle always equals 180 degrees regardless of the triangle's size. Measurements, such as parallax, depend on accurate angle assessment. A discrepancy can suggest properties like a triangle doubling back on itself or a potential measurement error.
Step-by-step explanation:
The question posed by the student involves understanding the relationship between the size of an angle in a triangle and the dimensions or shape of that triangle. According to the information provided, when Mr. Puncto observed that the discrepancy from 180 degrees in a triangle increased with the size of the triangle, it implies that there was an error in measurement or a misunderstanding of the geometric properties of triangles. In the study of geometry, the sum of the angles in any Euclidean triangle will always total 180 degrees, regardless of the size of the triangle. However, the shape of the triangle can indeed affect the measures of its angles.
For example, in the case of parallax measurements, the angle subtended by the baseline in triangles helps determine distances. A larger angle would suggest a closer object, while a smaller angle points to a more distant one. Additionally, under 20.3: A More Accurate Calculation, the zenith direction is mentioned, which impacts calculations when the assumption is that the Moon or any celestial body is directly overhead when it may not be. This reference helps us understand that the angles within a triangle, or measurements taken from them, can have significant effects depending on the assumptions made and the actual conditions.
In conclusion, if Mr. Puncto's measurements indicated increasing discrepancies from 180 degrees with increasing sizes of the triangles, it is possible that either the triangles in question doubled back on themselves or that there was some error in measurement. No triangle in a Euclidean plane should have angles that sum to any number other than 180 degrees.