Download Review Problems for the Final Exam Hyperbolic Geometry

Review Problems for the Final Exam
Hyperbolic Geometry
1. Suppose that ` ⊥ m and ` ⊥ n, m ⊥ t Prove that ` k t in Hyperbolic geometry. Then explain why
it is not guaranteed that t ⊥ n.
2. Prove that in Hyperbolic geometry the angles of a convex pentagon add up to less than 540 degrees.
3. Suppose that ` k m and A, B ∈ `, C, D ∈ m so that AC ⊥ m and BD ⊥ m. Show that if we are in
Euclidean Geometry that AC = BD but if we are in Hyperbolic geometry this might not be the case.
4. Suppose that it is true that the sum of angles in every triangle must be some number S. Prove
that S = 180.
5. Suppose that the angles in a triangle could add up to more than 180 degrees. Explain why there
might not be any parallel lines.
6.
Suppose that A, B and C are three noncollinear points so that there is no point D such that
AD = BD = CD (This is possible in Hyperbolic Geometry). Let ` be the perpendicular bisector of
AB and m be the perpendicular bisector of BC. Prove that ` k m.
7. Suppose that A, B, C, D, E are five points in hyperbolic geometry so that 4ABC and 4ADE are
triangles with B the midpoint of AD and C the midpoint of AE. Then note that AD = 2AB, AE = 2AC
and ∠BAC = ∠DAE. Prove that either ∠ABC ∠ADE or ∠ACB ∠AED (or both). (The best
way to prove this is to assume that both pairs of angles are congruent and then derive a contradiction).
“Definition” Problems
The final may contain a few short problems asking you to identify why something is not a valid model
of a set of axioms. You will not need to prove anything for these problems, only identify which axiom
is not satisfied.
8. Suppose that we have a geometry defined by the following axioms:
• There are four points.
• Every set of two lines intersect at a unique point.
• Every line has exactly two points.
Consider the model where points are corners of a square, and the lines are the edges of the square
together with the two diagonals. Why isn’t this a model for this geometry?
The next three examples will all use the incidence axioms as the axioms for the geometry:
• I-1: Each two distinct points determine a line.
• I-2: Three non-collinear points determine a plane.
• I-3: If two points lie in a plane, then any line containing those points also lies in the plane.
• I-4: If two planes intersect, then their intersection is a plane.
• I-5: Every line contains at least two points. Every plane contains at least three non-collinear
points. Space consists of at least four nonplanar points.
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9. Suppose that we define points to be the letters A, B, C, D, lines to be the sets {A, B}, {A, C}, {A, D}, {B, C}
and {C, D}, planes to be the sets {A, B, C} and {B, C, D} and space to be the set {A, B, C, D}. Why
doesn’t this model satisfy the incidence axioms?
10. Suppose that we define points to be points on a sphere of radius 1, lines to be diameters of the
sphere, the only plane to be the sphere and space to also be the sphere. Why doesn’t this model satisfy
the incidence axioms?
11.
Suppose that we define points to be any points on the Euclidean triangles 4ABC or 4BCD
or their interiors, lines to be intersections of Euclidean lines and the two two triangles, planes to be
the triangles 4ABC and 4BCD together with their interiors and space to be all points in these two
triangles. Why doesn’t this model satisfy the incidence axioms?
12. Recall the first three distance axioms:
• D-1: Each pair of points A and B has a number associated to it called the distance and indicated
by AB.
• D-2: For all points A and B, AB ≥ 0 with AB = 0 iff A = B.
• D-3: For all points A and B, AB = BA.
Suppose that we define distance in the following way: If A can be represented by the vector u and B can
be represented by the vector v then AB = u · v. Why doesn’t this model satisfy the distance axioms?
13. Again, suppose that we define distance by taking the Euclidean distance and rounding down to
make the number an integer. Why is this not a valid model for distance?
14. Recall the plane separation postulate:
Axiom 0.1 Let ` be any line lying in plane P . Then P − ` can be divided into two half planes H1 and
H2 so that:
• H1 ∩ H2 = ∅.
• If A and B both lie in the same half plane, then AB lies in that half plane.
• If A and B lie in different half planes, then AB intersects `.
Suppose that we have a model for geometry where we use Euclidean geometry, but only take the points
√
whose coordinates are all integers. (So for example (0, 0) and (2, −3) are points but (0, .5) and (π, 2)
are not). Lines are and lines segments are defined normally, but only include these integer coordinate
points.
Why does this model violate the plane separation axiom?
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Results of Absolute Geometry
Betweenness results may be assumed for the purposes of this exam.
−−→
• Segment construction theorem: Given a length L and a ray BC, it is possible to find a point X
−−→
on BC so that BX = L.
−→
←→
• Angle construction theorem: Given an angle measure M , a ray AB and a point C not on AB, it
←→
is possible to find a point X on the C-side of AB so that m∠XAB = M .
• Postulate of Pasch: If a line intersects the interior of AB in triangle 4ABC, then it intersects
exactly one of the following: the interior of BC, the interior of AC or C.
• Angle Pairs: Linear pairs of angles are supplementary and vertical pairs of angles are congruent.
• Congruence Criteria for Triangles: 4ABC ∼
= 4DEF iff there is one of the following congruence
patterns: SAS, ASA, SSS or AAS.
• SSA Theorem: If ∠ABC and ∠DEF are two triangles so that ∠BAC ∼
= DE and
= ∠EDF , AB ∼
∼
BC = EF , then the angles ∠BCA and ∠EF D are either congruent or supplementary.
• Isosceles Triangle Theorem: If two angles of a triangle are congruent, the opposite sides are also
congruent. If two sides of a triangle are congruent, the opposite angles are also congruent.
• Exterior Angle Theorem: In any triangle, the measure of an exterior angle is greater than the
measure of either corresponding opposite interior angle.
• Scalene Inequality: In any triangle, the larger angle is opposite the larger side and the larger side
is opposite the larger angle.
• Corollary of Scalene Inequality: The angles of an equilateral triangle are all congruent.
• Triangle Inequality: The length of a side of triangle is less than the sum of the length of the
remaining two sides.
• Saccheri-Legendre Theorem: The sum of the measures of the angles of a triangle is at most 180.
In Euclidean geometry the angles add up to exactly 180. In Hyperbolic geometry the angles add
up to less than 180.
• Convex Set Criterion: A set S is convex iff for every set of points A, B in S, AB ⊂ S.
• Congruence in right triangles: Two right triangles are congruent iff two of their sides are congruent.
• Convex Quadrilaterals: A quadrilateral is convex iff its diagonals intersect.
• Tangents and Secants: Suppose line ` intersects a circle in the same plane with center O at point
P . Then ` intersects the circle exactly once iff ` ⊥ OP and intersects the circle exactly twice
otherwise.
• Defect of a triangle: If 4ABC is a triangle and D is a point on the interior of BC, so that the
defect of 4ABC is d1 , the defect of 4ABD is d2 and the defect of 4ACD is d3 , then d1 = d2 + d3 .
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Results of Euclidean Geometry
• Playfair’s Axiom: Given a line ` and a point P there is exactly one line m such that m k `.
• Parallel Line Criteria: Suppose that `, m and t are all lines in the same plane with t the transversal
line. Then ` k m iff one of the following is true: alternating interior angles are congruent,
interior angles on the same side of the transversal are supplementary, or corresponding angles are
congruent.
• Euclidean Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the
sum of the measures of the two opposite interior angles.
• Angular Sum in a Convex Polygon: The sum of the measures of the angles in a convex n-gon add
up to (n − 2) · 180. In particular, the sum of angles in a triangle is 180.
• Side Splitting Theorem: Suppose that in 4ABC, E is an interior point of AB and F is an interior
point of AC. Then BC k EF iff
AE
AF
=
AB
AC
• AA Criterion for similarity: If two angles are congruent in two triangles, then the triangles are
similar.
• SSS Criterion for similarity: If between two triangles the ratio between corresponding sides is
always the same, then the triangles are similar (so the corresponding angles are congruent).
• SAS Criterion for similarity: If between two triangles the ratio between two corresponding sides
are the same, and the angle between those sides are congruent, then the triangles are similar.
• Results about Parallelograms: The diagonal of a parallelogram divides it into two congruent
triangles. In particular, the opposite sides of a parallelogram are congruent and the opposite
angles of a parallelogram are congruent.
• The Pythagorean Theorem: Suppose that we have a triangle with side lengths a, b, and c. Then
the triangle is a right triangle with hypotenuse the side of length c iff
a2 + b 2 = c 2
• The Inscribed Angle Theorem: Suppose that A, B, C are points on a circle, and D is a point on
the interior of ∠ABC so that ADC (so this is a minor arc). Then
_
_
1
m∠ABC = mADC
2
• Two Chord Theorem: If AB and CD are chords of the same circle which meet at point P , then:
AP · P B = CP · P D
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Later Results
• The only regular polygons which tile the plane are the equilateral triangle, square and regular
hexagon.
• Axioms of Area:
– To every “nice” region R in a plane (which includes the interiors of polygons and circles,
as well as all finite intersections and unions of these regions) there is a number Area(R)
associated with that region called its area.
– If R1 and R2 are regions in a plane so that R1 ⊆ R2 , then Area(R1 )≤Area(R2 )
– If R1 and R2 are regions in a plane so that Area(R1 ∩ R2 ) = 0, then Area(R1 ∪ R2 ) =
Area(R1 )+Area(R2 )
– If two regions are congruent, they have equal area. Interiors of congruent polygons are
congruent and interiors of circles with the same radius are congruent.
– The area of the unit square is 1.
– If R1 and R2 are two regions in the plane so that for every line ` parallel to some fixed line
n, ` ∩ R1 and ` ∩ R2 have the same length, then Area(R1 ) = Area(R2 )
• Results about vectors:
– u · u = ||u||2 .
– u · v = ||u|| · ||v|| cos(θ) where θ is the measure of the angle in between the two vectors.
– Two vectors u and v are parallel iff there is some nonzero λ so that u = λv.
– Two vectors are perpendicular iff u · v = 0.
– u + v is the diagonal of the parallelogram with sides u and v which lies in between the two
vectors.
• Results about Transformations:
– Definitions:
∗ A linear transformation is a one-to-one and onto map which sends lines to lines. It
preserves collinearity and betweenness.
∗ A similitude is a linear transformation with the property that if it sends A to A0 , B to
B 0 , then A0 B 0 = kAB. The number k is called the dilation factor. Similitudes preserve
angles. Similitudes send triangles to similar triangles.
∗ An isometry is a linear transformation with the property that if it sends A to A0 , B to
B 0 , then A0 B 0 = AB. Isometries send triangles to congruent triangles.
∗ A linear transformation is called direct if it preserves orientations and opposite if it
reverses orientations.
– The composition of any number of linear transformations is a linear transformation. The
composition of any number of similitudes is a similitude. The composition of any number
of isometries is an isometry. The composition of any number of direct transformations is
a direct transformation. The composition of an odd number of opposite transformations is
opposite and the composition of an even number of opposite transformations is direct.
– If an isometry fixes three noncollinear points, then it is the identity map.
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– Every isometry is the composition of at most three reflections.
– A transformation is a translation iff it is the composition of two reflections over parallel lines.
– A transformation is a rotation iff it is the composition of two intersecting lines. The point
of intersection is the center of rotation, and the angle of the rotation is twice the measure of
the smaller angle formed by the intersecting lines.
– The identity map fixes every point. A rotation or dilation fix only its center. A reflection
over a line fixes only the line. A translation does not fix any point.
– The inverse of a translation is the translation with equal length in the opposite direction.
The inverse of a rotation is the rotation of equal measure with the same center in the opposite
direction. The inverse of a reflection is the reflection. The inverse of a dilation is the dilation
with reciprocal dilation factor and the same center.
– Given two congruent triangles, there is a unique isometry which maps one onto the other.
Likewise, given two similar triangles, there is a unique isometry which maps one onto the
other.
– Coordinate Forms of Transformations
∗ A general linear transformation has the form:
x
a
b x
h
T
=
+
y
c
d y
k
where ad 6= bc.
∗ A general similitude has the form:
x
a
T
=
y
b
−δb
δa
x
h
+
y
k
where δ = ±1. The dilation factor is a2 + b2 . If a2 + b2 = 1 and δ = 1, then T is a direct
isometry. If a2 + b2 = 1 and δ = −1, then T is an opposite isometry.
∗ Dilations with center (0, 0) have the following form:
x
k
0 x
T
=
y
0
k y
∗ The rotation of θ degrees counterclockwise with center of (0, 0) has the form:
x
cos(θ)
− sin(θ) x
T
=
y
sin(θ)
cos(θ)
y
∗ The reflection over the line ax + by = 0 has the following form:
2
1
b − a2
−2ab
x
x
T
=
2
2
2
2
−2ab
a −b
y
y
a +b
∗ The translation which sends (0, 0) to (h, k) has the following form:
x
1
0 x
h
T
=
+
y
0
1 y
k
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