Why is the fourth angle of a Lambert quadrilateral obtuse in elliptic geometry?

Question

I know that fourth angles of Lambert quadrilaterals are acute in hyperbolic geometry and right in Euclidean, but why are fourth angles only able to be obtuse in elliptic geometry?

Edit: Some background information. We are learning about Lambert and Saccheri quadrilaterals. My professor mentioned that the fourth angles are right and acute in Euclidean and hyperbolic geometry, respectively. However, she did not mention elliptic geometry. I assume it has something to do with the fact that there are no "straight" lines in elliptic geometry, or that there are no parallel lines, but I am not sure. Am I on the right track?

Edit 2: Elliptic geometry is a non-Euclidean geometry with positive curvature which replaces the parallel postulate with the statement "through any point in the plane, there exist no lines parallel to a given line." https://mathworld.wolfram.com/EllipticGeometry.html

That is not a precise definition. You can't prove anything about elliptic geometry without a precise definition, and how you prove it depends on the details of the definition. — Eric Wofsey, Apr 28 '20 at 19:21

score 1 · Answer 1 · edited Feb 14 '22 at 16:18

Like in spherical trigonometry there is spherical excess, sum of three angles is in excess of two right-angles in elliptic geometry.

When two triangles are combined into a quadrilateral the sum of interior angles is in excess of four right-angles.

If three angles are forced to be right-angles then the fourth remaining angle is forced to be obtuse inside of the Lambert quadrilateral.

In hyperbolic geometry the left out angle would by the same token be acute.

Why is the fourth angle of a Lambert quadrilateral obtuse in elliptic geometry?

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