Download Review for Final Exam Hyperbolic Geometry Unified

Review for Final Exam
Hyperbolic Geometry
For these problems you can use any of the results from unified geometry, together with the results
derived from the Hyperbolic geometry. In particular you can assume that α = ∞, which simplifies
many results from unified geometry.
1. Suppose that A, B, C are points on a circle with center O so that AC is a diameter of the circle
and BO ⊥ AC. Show that the angle sum of 4ABC is less than the angle sum of 4AOB.
2. Suppose that A, B, C, D, E, F, G, H, I are all points on a circle with center O so that A, B, C,
D, E, F, G, H, I is a regular nonagon (nine sided polygon) and
AB = BC = CD = DE = EF = F G = GH = HI = IA = r
where r is the radius of the circle. Determine the measurement of ∠ABC. Set β = 180.
3. Suppose that ` ⊥ m, m ⊥ n and n ⊥ t. Show that ` k n and m k t. Then explain why t cannot be
perpendicular to `.
4.
Suppose that A, B, C are three noncollinear points so that there is no circle which contains all
three. That is, for every point O, the distances AO, BO and CO are not all equal to each other. Let `
be the perpendicular bisector of AB and m be the perpendicular bisector of BC. Prove that ` k m.
5. Suppose that A, B, C, D, E are five points so that 4ABC and 4ADE are triangles with B the
midpoint of AD and C the midpoint of AE. Prove that either ∠ABC is not congruent to ∠ADE
or ∠ACB is not congruent to ∠AED (which includes the possibility that neither pair of angles is
congruent).
6. Suppose that A, B, C, D, E are five points so that A, C, E are collinear, B, C, D are collinear and
4ABC and 4CDE are equilateral triangles. Show that 4ABC ∼
= 4CDE.
7. Suppose that P is a point not on `, X is the foot of P on ` and θ is the angle of parallelism at P
−→
relative to `. (That is, if A is a point so that m∠XP A < θ, then P A intersects ` and if A is a point so
−→
←→
that θ ≤ m∠XP A then P A does not intersect `). Furthermore, suppose that B is a point on P X so
−−→
that (BP X) is true and C is a point so that m∠XBC = θ. Show that BC does not intersect `.
8.
Suppose that 4ABC is an equilateral triangle, that M is the midpoint of AB, that N is the
midpoint of BC and O is the midpoint of CA. Show that 4M N O is an equilateral triangle. Then show
that m∠ABC < m∠M N O.
←→ ←→ ←→ ←→ ←→ ←→
9. Suppose that A, B, C, D are four points so that AB ⊥ BC, BC ⊥ CD, CD ⊥ DA and ♦ABCD
is a convex quadrilateral. Show that ∠DAB is acute.
10.
Suppose that A, B, C, D, E are five points so that B, C, D are collinear with (BCD) true, A
←→
and E are on the same side of BD, ∠ABC ∼
= ∠ECD ∼
= ∠CEA and ∠ABC ∼
= ∠EAC. Show that
4ABC 4CEA.
Unified Geometry
The problems for unified geometry will correspond to the problems from the first two exams. Therefore,
the best place to study these problems is the reviews for the first two exams. However, there are some
problems where it will be tempting to use Euclidean results. This will not be allowed. The problems
that follow should not be viewed as fully representative of the types of questions that I will ask about
unified geometry. The most comprehensive representation of the types of problems that I may ask can
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be found by looking at the reviews for the first two exams. Instead, these problems are the types of
problems that would be easiest to get wrong by referring to Euclidean results.
11. Suppose that A, B, C, D are four collinear points so that (ABC) is true and (ACD) is true. Prove
that there is a ray which contains all four points A, B, C and D.
12.
Suppose that A, B, C are three collinear points so that AB = 2, BC = 3 and so that the
statements (ACB), (CAB) are false. Find all possible values for AC.
13. Suppose that A, B, C are three collinear points so that (ABC) is true, and D, E are points not
on the same line so that 4DAB is an equilateral triangle and DE = BC, AB = EC. Furthermore,
suppose that m∠DEC = 130. Find the measure of m∠DAB. (Use β = 180).
←→
14. Suppose that A, B, C, D are four points so that A and D are on opposite sides of the line BC,
that m∠ABC = 62 = m∠BCD, m∠BAC = 61 = m∠CBD and m∠ACB = 60 = m∠CDB. Is
4ABC ∼
= 4BCD? All measurements are in degrees (so β = 180).
15. Suppose that h, k and r are three concurrent rays so that r is in between h and k. Show that
there is a line which intersects all of h, k and r at interior points.
16. Suppose that α > 10 (which includes the possibilities that α = ∞ or α < ∞) and that A, B, C
and D are four points so that (BCD) is true, ∠BAD is acute, AB = BC = CA = 3 and AD = 5.
What is the largest angle which appears in either the triangle ∠ABC or 4ACD? (If there is a tie for
largest angles, indicate that).
17.
Suppose that A, B, C, D √
are four points so that (BCD) is true, AC ⊥ BD, AC = BC = 3,
CD = 4, AD = 5 and AB = 3 2 ≈ 4.243. Which are the smallest angles which appear in 4ABC,
4ACD and 4BAD?
18. Suppose that A, B, C, D is a convex quadrilateral so that AB = BC = CD = DA. Show that
∠ABC ∼
= ∠CDA. You may assume that AB, BC, CD, DA, AC, BD < α/2.
19. Suppose that A, B, C, D are four points on a circle with center O so that ♦ABCD is a convex
quadrilateral, that AB is a diameter of the circle, and that AD = BC. Prove that DB = CA.
Euclidean Geometry
The problems for Euclidean geometry will be similar to those from the third exam. The review for the
third exam does a good job of showing what problems I may ask in Euclidean geometry, and since the
class did very well on the Euclidean geometry test, I do not see a need to include more problems on
that topic in this review.
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Axioms
0. A line is a proper subset of the universal set.
1. Each line is a set of points having at least two members.
2. There exists at least two points. Additionally, given two distinct points P and Q, there exists at
least one line which contains P and Q.
3. (Metric Axiom) To each pair of points (A, B), distinct or not, there corresponds a real number
AB called the distance from A to B which satisfies the properties:
(a) AB ≥ 0, and AB = 0 if and only if A = B.
(b) AB = BA.
4. If P and Q are distinct points so that P Q < α, then there is exactly one line which contains P
and Q.
5. If A, B, C and D are any four distinct collinear points such that (ABC) is true, then at least one
of the following is true: (DAB), (ADB), (BDC) or (BCD).
6. (Ruler Postulate for Rays) It is possible to assign each of the numbers in the interval 0 ≤ x ≤ α
−→
to any ray AB so that the end point has coordinate 0, and so that if C[c] and D[d] are points on
the ray that CD = |c − d|.
7. To every angle hk there corresponds a real number hk called the measure of hk which satisfies the
properties:
(a.) hk ≥ 0, and hk = 0 iff h = k.
(b.) hk = kh.
8. All straight angles have the same measure, and every angle has a measure less than or equal to
that of a straight angle.
9. (Plane Separation Postulate) There corresponds to each line ` two regions H1 and H2 such that:
(a) The sets `, H1 and H2 are disjoint, and their union is the universal set.
(b) H1 and H2 are nonempty convex sets.
(c) If A ∈ H1 , B ∈ H2 and s is a segment containing A and B, then s intersects `.
10. (Angle Addition Postulate) If point D lies in the interior point of D, or is an interior point of one
of the sides, then m∠ABD + m∠DBC = m∠ABC.
←→
11. (Angle Construction Postulate) Given any line AB and half plane H determined by that line, for
every real number r between 0 and 180 (including 0 and 180), there is exactly one ray AP in H
such that m∠P AB = r.
12. (SAS Postulate) If 4ABC and 4DEF are two triangles such that AB ∼
= DE, ∠ABC ∼
= ∠DEF
∼
and BC ∼
EF
then
4ABC
4DEF
.
=
=
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Useful Results
• The statement (ABC) is true exactly when A, B and C are collinear and AC = AB + BC.
• The statement (huk) is true exactly when h, u, and k are concurrent and hk = hu + uk.
• If two distinct lines intersect at points A and B, then AB = α. If α = ∞, then two distinct lines
can only intersect at one point.
←→
• It is possible to give coordinates in the range −α < x ≤ α to the line AB so that A has coordinate
0, B has a positive coordinate, and if C[c] and D[d] are points on the line, then:
CD = |c − d| if |c − d| ≤ α, CD = 2α − |c − d| otherwise
−→
• It is possible to give coordinates in the range −β < θ ≤ β to the rays concurrent to the ray AB
−→
←→
so that AB has coordinate 0, that the rays on one side of AB have positive coordinates, and if h,
−→
k are rays concurrent to AB with coordinates θh , θk respectively, then:
hk = |θh − θk | if |θh − θk | < β, hk = 2β − |θh − θk | otherwise
• If A[a], B[b] and C[c] are three points on the same ray, then (ABC) is true exactly when a < b < c
or c < b < a is true.
• If A, B, C and D are any four distinct points so that (ABC) and (ACD) are true, then (ABCD)
is also true.
• Every ray has a unique opposite ray. If h is a ray, h0 is its opposite ray and ` is any ray not
containing h, then the interior points of h and h0 lie on opposite sides of `.
• Two points A and B are on opposite sides of the line ` if and only if there is a point X on ` so
that (AXB) is true.
• If a line segment or a ray has an endpoint of `, and an interior point on the side H1 of `, then
every interior point of that object will be on that side of `.
• If ` is a line, and A, B, C are distinct points not on `, then ` contains a point X so that (AXB) is
true, then either ` contains a point F so that (AF C) is true or a point G so that (BGC) is true.
• If ∠BAC is a nondegenerate, nonstraight angle, and D ∈ Int ∠BAC, then there is a pointX on
−−→
AD so that (BXC) is true.
• Linear pairs are supplementary. Vertical pairs are congruent.
• If ∠1 is supplementary to ∠2 and ∠2 is supplementary to ∠3, then ∠1 and ∠3 are congruent.
• If α < ∞, then given a point A, there is exactly one point A∗ such that AA∗ = α. Additionally,
if A ∈ `, then A∗ ∈ `.
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Results about Triangles, Polygons and Circles
• If two angles of a triangle are congruent, then their opposite sides are congruent, and conversely
if two sides of a triangle are congruent, then their opposite angles are congruent.
• If any two triangles there is a correspondence pattern of ASA or SSS, then the two triangles are
congruent under that correspondence.
• The perpendicular bisector of a line segment is exactly the set of points which are equidistant
from both endpoints of the circle.
←→
• The foot of a point P on a line ` (where P ∈
/ `) is the point F on ` such that P F ⊥ `. The
distance of P to ` is P F . If P F < α/2, then the foot is unique.
• If A, B and C are any three distinct points, then AB + BC ≤ AC, and AB + BC = AC if and
only if (ABC) is true.
• In a single triangle, the longer side is opposite the bigger angle and vice versa.
• If 4ABC and 4DEF are triangles such that AB ∼
= DE and BC ∼
= EF but m∠ABC > m∠DEF
then AC > DF .
• If 4ABC and 4DEF are triangles such that AB ∼
= DE, BC ∼
= EF and ∠BCA ∼
= ∠EF D, then
∼
either 4ABC = 4DEF or the angles ∠CAB and ∠F ED are supplementary.
• If 4ABC is a triangle with side lengths less than α/2, then every exterior angle is bigger than its
corresponding opposite interior angles.
• If 4ABC is a triangle with side lengths less than α/2, then the sum of the measures of any two
of its angles is less than β.
• If 4ABC and 4DEF are two triangles with side lengths less than α/2 so that ∠ABC ∼
= ∠DEF ,
∼
∼
∼
∠BCA = ∠EF D and AC = DF then 4ABC = 4DEF .
• If two right triangles with side lengths less than α/2 and which have a congruent pair of legs and
congruent hypotenuses, then the two triangles are congruent.
←→
←→
• If O is a point so that OC ⊥ AB with (ACB) and OA, OC, OB < α/2, then OA > OC.
Furthermore if X is a point so that (XAC) is true then OX > OA > OC and if Y is a point so
that (AXC) is true then OA > OX > CO.
• If a line contains an interior point of a circle, then it is secant to that circle (and so intersects it
at exactly two points).
• If a line is tangent to a circle, then it does not contain an interior point of the circle.
• If a line ` contains a point X on a circle with center O, then ` is tangent to the circle if and only
←→
if ` ⊥ OX.
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Results from Euclidean Geometry
All results on this page are only valid in Euclidean Geometry (that is, not in Hyperbolic or Spherical
Geometry)
• Euclidean Axiom: α = ∞. Furthermore, if ` is a line and P is a point not on `, then there is
exactly one line m which contains P and which is parallel to `.
• If ` k m and t is a transversal, then the corresponding angles are congruent, the alternate interior
angles are congruent, and the interior angles on the same side are supplementary.
• The angle sum of any triangle is exactly equal to β (180 degrees)
• The measure of an exterior angle is exactly equal to the sum of the opposite interior angles.
• The opposite sides of a parallelogram are congruent. The opposite angles of a parallelogram are
congruent.
←→ ←→
• If X is the midpoint of AB and Y is the midpoint of BC, then XY k BC and XY = (1/2)BC.
• In a triangle 4ABC the midpoint X of AB is equidistant from A, B and C iff ∠ACB is a right
angle.
• Given three points, there is a circle which contains all three.
• AA, SAS and SSS (where S indicates that the corresponding sides have the same ratio) are all
valid similarity criteria.
• In a triangle 4ABC AC 2 + CB 2 = AB 2 iff ∠ACB is right.
• If A, B, C are collinear and P is any point, then:
AB ∗ (P C ∗ )2 + BC ∗ (P A∗ )2 + CA∗ (P B ∗ )2 + AB ∗ BC ∗ CA∗ = 0
←→
• If P A is tangent to a circle with center r and radius O at point A, then:
P O 2 − r 2 = P A2
←→
• If AB is secant to a circle with center r and radius O at points A and B, then:
(P O∗ )2 − r2 = P A∗ · P B ∗
• If two angles inscribed in the same circle intercept the same arc, they are congruent.
• If two angles inscribed in the same circle intercept opposite arcs on that circle, they are supplementary.
←→
• If C and D lie on the same side of AB and ∠ACB ∼
= ∠ADB, then there is a circle which contains
A, B, C and D.
• An angle in a circle is right iff it intercepts the diameter of the circle.
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Results from Hyperbolic Geometry
The following results are valid only in Hyperbolic Geometry.
• Hyperbolic Axiom: α = ∞. Furthermore, if ` is a line and P is a point not on `, then there is
more than one line m which contains P and is parallel to `.
• The angle sum of any triangle is less than β (180 degrees). The difference between β and the
angle sum of a triangle 4ABC is called the defect of the triangle.
• The exterior angle of a triangle has measure more than the sum of the measures of the opposite
interior angles.
• If (BDC) is true, and 4ABC is a triangle, then the defect of 4ABC is the sum of the defects of
4ABD and 4ADC.
• The defect of a triangle can take on any value strictly between 0 and β.
• Given a line ` and a point P not on ` so that X is the foot of P on `, there is an angle measure
−→
−→
0 < θ < β/2 so that if 0 ≤ m∠XP A < θ, then P A intersects ` and if θ ≤ m∠XP A then P A does
not intersect `.
Facts about Spherical Geometry
These facts may be helpful, even though there are not any problems specifically about spherical geometry:
• α < ∞.
• If ` and m are distinct lines, then they intersect at two points X and Y so that XY = α.
• The angle sum of a triangle is always more than β (180 degrees)
• For every point A, there is exactly one point A∗ so that AA∗ = α.
• If AA∗ = α and B is any third point, then (ABA∗ ) is true.
• If A, B, C are collinear points, then (ABC), (BCA), (CAB) or (AB ∗ C) is true, where B ∗ is the
point so that BB ∗ = α.
• The exterior angle inequality and AAS congruence are only valid for triangles whose sides are of
length less than α/2.
• It is possible for a point P to have more than one foot on `, but only if the distance from P to
any point X on ` is α/2 (In this case, every point on ` is a foot of P ).
• It is possible for a triangle to contain two right angles, or a right angle and an obtuse angle.
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