Review for Exam One
Unless a problem specifies otherwise, you are free to use any axiom or any result proved from those
axioms. You are not restricted to the results from the last pages, but it may help to focus your arguments
using those results to make sure that you haven’t accidentally used something unproven.
1. For each of the following statements, determine whether it is:
(a.) Always true regardless of what α is.
(b.) Always true when α < ∞, but possibly false when α = ∞.
(c.) Always true when α = ∞, but possibly false when α < ∞.
(d.) Possibly false regardless of what α is.
A. If ` and m are distinct lines, then they intersect at at most one point.
B. If A[a], B[b] and C[c] are points on the line ` (with corresponding coordinates) such that a < b < c,
then (ABC) is true.
C. If h, k, u are three concurrent rays, then one of (hku), (huk) or (kuh) must be true. That is, either
hk + ku = hu, hu + uk = hk or ku + uh = kh.
D. If A, B and C are three collinear points, then one of (ABC), (BAC) or (ACB) must be true.
E. If ` and m intersect at three distinct points, then ` = m.
−→ ←→
F. If r is a ray with endpoint A and interior point B, then r = AB ⊂ AB.
G. If r is a ray and A is an endpoint of the ray and B is an interior point of the ray, then it is possible
←→
to find a point C on AB so that (ABC) is true and C is not on r.
H. For any given line `, there are points with coordinates (in the sense of Theorem 1.8.5) A[a], B[b],
C[c] so that a < b < c and there is a ray which contains A and B but not C.
I. Given a point A on `, there is a point A∗ on ` so that (ABA∗ ) is true for every other point B on `.
J. If hk and ku are angles, then the interior points of k lie in the interior of the angle hu.
−→
−→ −→
K. If AB is well defined, then AB ∪ BA is a line.
L. It is possible to find a line segment which has two endpoints but no interior points.
M. Every ray has two endpoints.
2. Using only Axioms 0-2, prove that there must be at least three points.
3. Prove that every ray is a convex set.
4. Prove that if AB < α, there is no ray r such that AB = r.
5. Let A and B be any two points on `. Prove that there are infinitely many points between A and
B on `. (Or equivalently, that there are not finitely many points between A and B on `)
6. Suppose that you are told that the four points, A, B, C and D are collinear, that (ABD) is true,
that the least upper bound for distance α = ∞, and that:
AB = 3, BD = 6, CD = 2
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Given this information, there are two possibilities for the value of AC. Find them.
7. Suppose that h, k, u are concurrent rays with ` the ray containing u. Furthermore, suppose that h
and k are on opposite sides of `. Let h0 be the opposite ray of h and u0 be the opposite ray of u. Prove
that either (ukh0 ) is true, (u0 kh0 ) is true, or h = k.
8. Prove that if AB = CD and A = C, then B = D. (Argue using the definition that every interior
point X of a line segment EF must satisfy (EXF ).)
9. Suppose that we have a model satisfying axioms 0-4 where A, B, C and D are points on the line
` so that:
AB = 8, AC = 4, AD = 10, BC = 4, BD = 2, CD = 7
List all betweenness relations between these points.
10. Suppose that α = 10, and that A, B and C are three collinear points so that AB = 6, AC = 9.
There are two possible values of BC. Find them, and justify your answer.
11. Suppose that A, B, C and D are points so that D ∈ Int ∠BAC, and so that
m∠BAC = 5 · (m∠BAD)
Prove that 4 · (m∠DAC) = m∠BAD.
12. Suppose that A, B and C are points so that (ABC) is true, that AB = 1 and that AC = 1/BC.
Prove that:
√
1+ 5
AC =
2
13. Suppose that A, B and C are three points so that (ABC) is true. Let O be any point which is
−−→
−→
−→
not collinear to A, B and C. Show that the ray OB is in between the rays OA and OC (in the sense
that m∠AOC = m∠AOB + m∠BOC).
14. Prove that if (ABC) and (ACD) are true, then there is a ray which contains the points A, B, C
and D. Justify why each point is on the ray.
−→ −→
15. Suppose that A, B are two points such that AB < α. Prove that AB ∩ BA is a line segment.
16. Suppose that A, B and C are three points so that (ABC) is true and AB = 2BC. Let P be the
midpoint of AB. Prove that P C = 2AP .
←→
←→
17. Suppose that the lines AB and CD intersect at point E, that (AEB) and (CED) are true, and
−→
that the ray EF is the bisector of ∠AED. Suppose that β = 180 and m∠AEC = 110. Find m∠DEF .
18. Suppose that h, u, k are distinct concurrent rays, and that the interior points of u and k are on
the same side of the line containing h. Prove that either (huk) or else hu + uk > hk.
19. Suppose that we measure angles in degrees (so that β = 180), and that r, s, t and u are concurrent
rays with (rst) and (stu) both being true. Furthermore, suppose that:
rs = st = tu = 70
What is ru?
20. Suppose that β = 180 and h, k and u are concurrent rays so that the angles hk = ku = 45. Prove
that either h ⊥ u (that is, hu = 90) or else h = u.
21. Prove that the interior of a non-straight (but possibly degenerate) angle is a convex set, and that
the interior of a straight angle is not a convex set.
22.
Suppose that A, B and C are three points which are not collinear, that ` contains a point on
Int ∠ABC, a point on Int ∠BAC and a point on AB (possibly an endpoint). Prove that either ` contains
a point on the interior of AC, or ` contains a point on the interior of BC or ` contains C.
23. Suppose that ∠1 and ∠2 are complementary, ∠2 and ∠3 are supplementary and ∠1 and ∠3 are
supplementary. Find the measures of the three angles.
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24. Suppose that h, k and u are three concurrent rays, that u0 is the opposite ray of u and that (hu0 k)
is true. Prove that hk + ku + uh = 2β.
25. Suppose that h and h0 are opposite rays, and k is any ray not on the line h ∪ h0 . Let a be the
bisector of the angle hk and b be the bisector of the angle kh0 . Prove that ab is a right angle.
26.
Let ` and m be lines, H1 a half plane determined by ` and I1 a half plane determined by m.
Prove that the intersection of H1 and H2 is the interior of some angle (even if H1 and I1 have an empty
intersection).
27. Suppose that β = 180, h, k and u are three concurrent rays so that hk = 30, ku = 80. There are
two possibilities for the value of hu. Find them (and justify your calculations).
28. Suppose that A, B and C are points so that (ABC) is true. Directly show that every point on
←→
−→
−−→
AB is either on BA or BC.
29. Show that if α = ∞ and A, B and C are collinear points, that AB + BC ≥ AC. (Note that we
have not yet shown that the triangle inequality is generally true and in fact lack an axiom for doing so!)
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Axioms
0. A line is a proper subset of the universal set.
1. Each 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.
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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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