Wikipedia:Reference desk/Archives/Mathematics/2024 August 15

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August 15

Can two polygons have the same interior angles (in order) if they are noncongruent and have no parallel sides?

Was thinking about the question of whether the interior angles, in order, determine a polygon up to congruence, when I realized that the answer is obviously no if there are parallel sides which you can contract and lengthen at will without changing any of the angles. Barring these polygons with parallel sides and their lengthenings/contractions, then, are there any polygons that share the same interior angles in order without being congruent? GalacticShoe (talk) 17:30, 15 August 2024 (UTC)

I'm not exactly sure what you're asking, but it seems to me that an example can be obtained by starting with a regular pentagon, and lengthening two adjacent sides, and the third opposite side, and shrinking the remaining two sides, while maintaining all angles. Tito Omburo (talk) 17:37, 15 August 2024 (UTC)

I'm realizing now that this question is embarrassingly simple; if we consider extending polygon sides to lines, with the sides then being line segments between the intersections of adjacent sides, then moving lines always preserves angles. For example, in the pentagon example, the lengthening/shrinking you mentioned is akin to moving two lines further away from the center of the pentagon. GalacticShoe (talk) 17:48, 15 August 2024 (UTC)

Upon further searching online, the answer is very much no yes; see this Math StackExchange post. In particular, there are many polygons where you can simply intersect the polygon with a planar half-space such that 1. the line L demarcating the planar half-space is parallel to one of the sides S, and 2. all sides of the polygon either fall on the interior of the half-space, share a vertex with S and intersect L, or are S. GalacticShoe (talk) 17:39, 15 August 2024 (UTC)

I think you meant to say that the answer is, Yes – they can have the same interior angles and yet be noncongruent (from tetragons on).  --Lambiam 22:25, 15 August 2024 (UTC)

Yup, silly mistake on my part; amended GalacticShoe (talk) 03:51, 16 August 2024 (UTC)