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Amanda Lutje
Math 329
October 7, 2015
Homework 7
1. Prove that the summit angles in a Saccheri quadrilateral must be congruent in neutral geometry.
Saccheri quadrilaterals are certain quadrilaterals whose base angles are right angles and
whose base-adjacent sides are congruent to one another. We can draw in two diagonals, Line BD
and Line AC because of I-1. Triangle DAB is congruent to Triangle CAB by C-6 (SAS) since we were
given that they have a congruent side, each have a right angle, and they share a base which is
congruent to itself by C-2. Since the two triangles are congruent, Line AC is congruent to Line DB.
We also have that Line AD is congruent to Line BC (given) and Line DC is congruent to Line DC
because of C-2. Then we have that Triangle BCD and Triangle CDA are congruent by definition 1.2 of
congruence, i.e. all the sides are the same length. Therefore Angle C and Angle D are congruent since
they are corresponding angles in congruent triangles.
Figure for proof:
2. The summit angles of a Saccheri quadrilateral cannot be obtuse because the interior angles of the
quadrilateral would then add up to be greater than 360. The interior angles of any quadrilateral
must add up to 360 or less in neutral geometry. A proof of this is found at this link:
https://faculty.etsu.edu/gardnerr/noneuclidean/hyperbolic.pdf
In doing further research, all proofs of this use the same fact in different ways to prove it.
3. If the summit angles in a Saccheri quadrilateral are acute, then the result is that the interior
angles will add up to less than 360. The summit line will not be equal in length to the base line.
Bisecting through the summit and base will give you two Lambert quadrilaterals. If the summit
angles are right angles, then you have what we have learned to be a rectangle in Euclidean
geometry. The summit and base will have the same length, drawing diagonals will give you several
congruent triangles, and the base-adjacent sides are parallel.
Examples of each:
4. Two types of Lambert squares (acute, right):
All Lambert squares have 3 right angles, and the fourth angle is up for debate. In Euclidean
geometry, the fourth angle would be a right angle and follow the definition of a quadrilateral as we
know it. In non-Euclidean geometries, the angle can be acute. My perception of these quadrilaterals
is different now that we have analyzed other types of geometries. The existence of these figures is
somewhat trivial if you are only thinking about Euclidean geometry, because they will only appear
to be quadrilaterals or squares with straight lines and right angles. Euclidean geometry makes it
easy to teach students the concepts of proofs and forming reasonable arguments in geometry. Many
complications and caveats arise when the axioms in Euclidean are not true, so I can see why it is not
introduced to younger students. I believe that students would benefit from exploring different types
and histories of geometry so they understand the significance of the Euclidean postulates that they
are learning.