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Test 1 - Axiomatic Geometry (Spring 2015)
INSTRUCTIONS: Complete each of the following problems in your Bluebook. Each problem
is worth a maximum of 15 points. Points will be awarded for both completeness and clarity
of solutions.
1. Let A denote the axiom system consisting of all the axioms of incidence geometry, together with the additional statement “There exist only finitely many points.” Let P denote
the statement “There are more than twice as many lines as there are points.” Prove that P
is independent of A.
2. Prove that if A and B are any two distinct points, there exists a unique point C such
that A ∗ C ∗ B and AC = 32 BC.
→
− −
−
3. Prove the Linear Triple Theorem: If ∠ab, ∠bc, and ∠cd are angles such that →
a ∗ b ∗→
c
→
− →
→
−
→
−
−
→
−
◦
and b ∗ c ∗ d , and a and d are opposite rays, then m∠ab + m∠bc + m∠cd = 180 .
4. Prove Pasch’s Theorem: If 4ABC is a triangle and ` is a line which does not contain
A, B, or C, and ` intersects AB, then either ` intersects AC or ` intersects BC. (Hint: To
prove this, you only need one postulate of neutral geometry.)
BONUS. (+8 points) Let ` be a line, and let A and B be distinct points on `. Suppose f : ` → R and g : ` → R are both coordinate functions, and suppose they satisfy
f (B) − f (A) = g(B) − g(A). Prove that there exists a constant K for which
g(X) = f (X) + K,
for all X ∈ `. (Hint: What should the constant K be?)
Axiom Systems
Incidence Geometry Axiom 1. There exist at least three distinct noncollinear points.
Incidence Geometry Axiom 2. Given any two distinct points, there is at least one line
that contains both of them.
Incidence Geometry Axiom 3. Given any two distinct points, there is at most one line
that contains both of them.
Incidence Geometry Axiom 4. Given any line, there are at least two distinct points that
lie on it.
Neutral Geometry Postulate 1 (Set Postulate). Every line is a set of points, and there
is a set of all points called the plane.
Neutral Geometry Postulate 2 (Existence Postulate). There at exist at least three distinct noncollinear points.
Neutral Geometry Postulate 3 (Unique Line Postulate). Given any two distinct points,
there is a unique line containing both of them.
Neutral Geometry Postulate 4 (Distance Postulate). For every pair of points A and B,
the distance from A to B is a nonnegative real number determined by A and B.
Neutral Geometry Postulate 5 (Ruler Postulate). For every line `, there is a bijection
f : ` → R with the property that for any two points A, B ∈ `, we have AB = |f (B) − f (A)|.
Neutral Geometry Postulate 6 (Plane Separation Postulate). For any line `, the set of
all points not on ` is the disjoint union of two subsets of the plane called the sides of `. If
A and B are two distinct points not on `, then A and B are on the same side of ` if and
only if AB ∩ ` = ∅.
Neutral Geometry Postulate 7 (Angle Measure Postulate). For every angle ∠ab, the
measure of ∠ab is a real number lying in the closed interval [0, 180] determined by ∠ab.
−
Neutral Geometry Postulate 8 (Protractor Postulate). For every ray →
r and every point
←
→
→
−
→
−
P not on the line r containing r , there is a bijection g : HR( r , P ) → [0, 180] which
→
−
−
−
−
−
−
satisfies: (1) g(→
r ) = 0; (2) if →
s is opposite to →
r then g(→
s ) = 180; and (3) if →
a and b
→
−
−
−
are any two rays in HR(→
r , P ) then m∠ab = |g( b ) − g(→
a )|.
Neutral Geometry Postulate 9 (SAS Postulate). If there is a correspondence between
the vertices of two triangles such that two sides and the included angle of one triangle are
congruent to the corresponding sides and angle of the other triangle, then the triangles are
congruent under that correspondence.