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Math/MthEd 362 Exam 2
Section 1, Dr. Siebert
1. For each statement below, determine whether it is true for only Euclidean (E) geometry, for only
hyperbolic (H) geometry, for both (B) types of geometry, or for neither (N) type of geometry. Put the
appropriate letter in the space to the left of each statement.
____ a. A right triangle exists.
____ b. Exactly one rectangle exists.
____ c. SAS holds.
____ d. A Lambert quadrilateral exists.
____ e. A triangle with defect greater than 0 exists.
____ f.
The Hinge Theorem holds.
____ g. It is possible to construct a triangle similar to another triangle.
____ h. There are some convex quadrilaterals with a defect of 0 and some convex quadrilaterals with a
defect greater than 0.
____ i.
Given any two parallel lines l and m, if a third line t intersects l at a single point, then t will
also intersect m.
____ j.
There exist lines l, m, and n such that l || m and l ⊥ n, but m and n are neither parallel nor
perpendicular to each other.
2. Does the double perpendicular construction imply the Euclidean parallel postulate? Justify your
answer.
3. Draw a picture that illustrates Euclid's Postulate V. Briefly explain why your picture illustrates this
postulate.
4. Suppose that 𝐴𝐵 is the longest side of the triangle ABC. Will the foot of the perpendicular from C
to 𝐴𝐵 lie in the segment AB? If so, prove it. If not, provide a counterexample.
5. Name the axiom, postulate, or theorem that allows you to do the following (note: there may be more
than one correct answer, but you only need to write one of them):
a. Conclude that a triangle has two angles that are congruent.
b. Conclude that the longest side of a triangle is the one that is across from the angle with the
greatest measure.
c. Make conclusions about interior angles if two lines are parallel.
d. Conclude that the Elliptic Parallel Postulate is not consistent with the axioms and theorems of
neutral geometry.
e. Guarantees that linear pairs are supplementary.
6. What must you prove to conclude… (Note: there may be more than one correct answer, but you only
need to write one.)
a. A quadrilateral is convex?
b. Two right triangles are congruent?
c. A point lies on the angle bisector?
e. Two lines are parallel?
d. The Euclidean Parallel Postulate holds?
7. Let ABCD be a quadrilateral such that 𝐴𝐵 ≅ 𝐶𝐷 and 𝐵𝐶 ≅ 𝐴𝐷. Prove that ABCD is a
parallelogram.
8. Suppose that we had chosen ASA instead of SAS as the final axiom for neutral plane geometry. Give
a proof of SAS using only ASA and no other triangle congruence theorems (e.g., AAS, SSS).
Side-Angle-Side Theorem: If ABC and DEF are two triangles such that 𝐴𝐵 ≅ 𝐷𝐸,
∠𝐴𝐵𝐶 ≅ ∠𝐷𝐸𝐹, and 𝐵𝐶 ≅ 𝐸𝐹, then ABC ≅ DEF.