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Non-Euclidean Geometry Exercises
1. A Lambert quadrilateral has three right angles. Prove that, in hyperbolic geometry, a Lambert
quadrilateral’s fourth angle (angle x) is acute (i.e., is less than one right angle).
x
2. A rhombus is a quadrilateral with equal sides. Suppose ABCD is a Lambert quadrilateral
in hyperbolic geometry. Prove that ABCD is not a rhombus. (Hint: Prove by reductio ad
absurdum. Assume for the sake of argument that ABCD is a rhombus, and show that this
assumption leads to a contradiction concerning what we know to be true about triangles in
hyperbolic geometry.)
3. Prove that, in elliptic (i.e., Riemmanian) geometry, the sum of the interior angles of any quadrilateral is more than four right angles.
4. A Saccheri quadrilateral has two equal sides, each perpendicular to a common base. Prove that,
in elliptic geometry, the summit angles (α and β) of a Saccheri quadrilateral are obtuse.
α
β