Editing Triangle (section)

=== Triangle in non-planar space ===
{{main article|Hyperbolic triangle|Spherical triangle}}
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A non-planar triangle is a triangle not embedded in a [[Euclidean space]], roughly speaking a flat space. This means triangles may also be discovered in several spaces, as in [[hyperbolic space]] and [[spherical geometry]]. A triangle in hyperbolic space is called a [[hyperbolic triangle]], and it can be obtained by drawing on a negatively curved surface, such as a [[saddle surface]]. Likewise, a triangle in spherical geometry is called a [[spherical triangle]], and it can be obtained by drawing on a positively curved surface such as a [[sphere]].{{sfn|Nielsen|2021|p=[https://books.google.com/books?id=hHMjEAAAQBAJ&pg=PA154 154]}}

The triangles in both spaces have properties different from the triangles in Euclidean space. For example, as mentioned above, the internal angles of a triangle in Euclidean space always add up to 180°. However, the sum of the internal angles of a hyperbolic triangle is less than 180°, and for any spherical triangle, the sum is more than 180°.{{sfn|Nielsen|2021|p=[https://books.google.com/books?id=hHMjEAAAQBAJ&pg=PA154 154]}} In particular, it is possible to draw a triangle on a sphere such that the measure of each of its internal angles equals 90°, adding up to a total of 270°. By [[Girard's theorem]], the sum of the angles of a triangle on a sphere is <math> 180^\circ \times (1 + 4f) </math>, where <math> f </math> is the fraction of the sphere's area enclosed by the triangle.<ref>{{cite web|last=Polking |first=John C. |url=https://www.math.csi.cuny.edu/~ikofman/Polking/gos4.html |title=The area of a spherical triangle. Girard's Theorem. |website=Geometry of the Sphere |access-date=2024-08-19 |date=1999-04-25}}</ref><ref>{{cite web|last=Wood |first=John |url=https://homepages.math.uic.edu/~jwood/freshsem/ |title= LAS 100 — Freshman Seminar — Fall 1996: Reasoning with shape and quantity |access-date=2024-08-19}}</ref>

In more general spaces, there are [[comparison theorem]]s relating the properties of a triangle in the space to properties of a corresponding triangle in a model space like hyperbolic or elliptic space.{{sfn|Berger|2002|pp=134–139}} For example, a [[CAT(k) space]] is characterized by such comparisons.{{sfn|Ballmann|1995|p=viii+112}}