Download Last year`s exam questions

MAT402H5F page 3 of 12
(1) For each of the following statements, specify whether it is true or false, and,
in one sentence, explain why.
(a) The French mathematician Legendre proved the parallel postulate.
(b) The sides of a Saccheri quadrilateral are divergently parallel lines.
(c) If AB < CD then there exists a point E between C and D such that
AB ∼
= CE.
(d) It is impossible to prove in neutral geometry that parallel lines exist.
continued on page 4
[12 points]
MAT402H5F page 4 of 12
[12 points]
(2) For each of the following statements, specify whether it is true or false, and,
in one sentence, explain why.
(a) In hyperbolic geometry, if two parallel lines have a common perpendicular, then they are equidistant.
(b) The alternate interior angle theorem in neutral geometry states that
if parallel lines are cut by a transversal then alternate interior angles
are congruent to each other.
(c) The first two mathematicians who published accounts of hyperbolic
geometry were Nikolai Ivanovich Lobachevsky and János Bolyai.
(d) Given two non-parallel planes in R3 and a point outside them, central
projection with respect to the point takes parallel lines on the first
plane to parallel lines on the second plane.
continued on page 5
MAT402H5F page 5 of 12
(3) For each of the following statements, specify whether it is true or false, and,
in one sentence, explain why.
(a) In hyperbolic geometry, if 4ABC and 4DEF are equilateral triangles
and A ∼
= D, then the triangles are congruent.
(b) In hyperbolic geometry, opposite sides of any parallelogram are congruent to each other.
(c) If we add to the axioms of neutral geometry the elliptic parallel postulate (that no parallel lines exist) then we get another consistent
geometry called elliptic geometry.
(d) The transformation z 7→ iz − i is an h-transformation (hyperbolic
transformation) of the Poincaré upper half-plane model.
continued on page 6
[12 points]
MAT402H5F page 6 of 12
[10 points]
(4) Give the definition of each of the following terms.
(a) A right angle.
(b) A Lambert quadrilateral.
(c) The interior of BAC.
(d) A rectangle.
(e) In the Poincaré upper half-plane, h-segments AB and DE being congruent.
continued on page 7
MAT402H5F page 7 of 12
(5) Give the statement of each of the following assertions.
(a) Transitivity of parallelism.
(b) The hyperbolic axiom.
(c) Clairaut’s axiom.
(d) The crossbar theorem.
(e) Archimedes’s axiom.
continued on page 8
[10 points]
MAT402H5F page 8 of 12
[6 points]
(6) For each of the following terms, draw an example in the Klein disk model:
(a) Supplementary angles.
(b) Asymptotic parallel lines.
(c) Divergent parallel lines.
continued on page 9
MAT402H5F page 9 of 12
(7) For each of the following terms, draw two different examples in the
Poincaré upper half-plane model.
(a) an angle
(b) side of a line
(c) counterexample to Hilbert’s parallel postulate.
continued on page 10
[9 points]
MAT402H5F page 10 of 12
[10 points]
(8) Justify each step of the following proof that supplements of congruent angles
are congruent:
Given: ABC ∼
= DEF .
To prove: CBG ∼
= F EH, where A ? B ? G and D ? E ? H.
(a) The points A, C, and G being given arbitrarily on the sides of ABC
and of its supplement, CBG, we can choose the points D, F , and
H on the sides of the other angle and of its supplement in such a way
that AB ∼
= EH.
= F E, and BG ∼
= DE, CB ∼
(b) With this choice, 4ABC ∼
= 4DEF .
(c) Hence, AC ∼
= DF and A ∼
= D.
(d) Also, AG ∼
= DH.
(e) Hence, 4ACG ∼
= 4DF H.
(f) Therefore, CG ∼
= F H.
(g) Hence, 4CBG ∼
= 4F EH.
(h) It follows that CBG ∼
= F EH, as desired.
continued on page 11
MAT402H5F page 11 of 12
(9) Prove that the following assertion implies Hilbert’s parallel postulate: “if a
line intersects one of two parallel lines and is distinct from it, then it also
intersects the other”.
continued on page 12
[10 points]
MAT402H5F page 12 of 12
[9 points]
(10) Show that the incidence axiom I-1 holds in the Poincaré upper half-plane
model.