Question

Prove in hyperbolic geometry: If two Saccheri quadrilaterals have congruent bases and congruent summits, then they are congruent.

Answer #1

Describe the Saccheri Quadrilateral and prove in neutral
geometry that the summit angles are congruent.

show that if the summit and the base angles of two saccheri
quadrilaterals are congruent then the reamining sides are
congruent.

prove that rectangles dont exist in hyperbolic geometry

formally/step by step: prove that similar but non-congruent
triangles cannot exist in elliptic geometry

Prove: If two angles of a triangle are not congruent, then the
sides opposite those angles are not congruent.

Prove that equilateral triangles exist in
neutral geometry (that is, describe a construction
that will yield an equilateral triangle). note that all the
interior angles of an equilateral triangle will be congruent, but
you don’t know that the measures of those interior angles is
60◦.Also, not allowed to use circles.

Draw an example of two non-congruent triangles that
have congruent corresponding angles and two pairs of sides (not
corresponding) congruent. If not, explain why.

Prove that for any integer a, a^37 is congruent to a
(mod 1729).
We have Fermat's Little Theorem and Euler's Theorem to
work with.

Neutral Geometry Proof
If two parallel lines are cut by a transversal, then
The alternate interior angles are congruent.
The corresponding angles are congruent.
The interior angles on the same side are supplementary.

Neutral Geometry Proof:
If two lines are cut by a transversal and
the alternate interior angles are congruent, or
the corresponding angle are congruent, or
the interior angles on the same side are supplementary, then the
lines are parallel.

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