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Geometry: Neutral
MATH 3120, Spring 2016
Many theorems of geometry are true regardless of which parallel postulate is used. A neutral geometry is one in which no parallel postulate exists, and the theorems of a netural geometry are true for
Euclidean and (most) non-Euclidean geomteries.
Spherical geometry is a special case of Non-Euclidean geometries where the great circles on the sphere
are lines. This leads to spherical trigonometry where triangles have angle measure sums greater than
180◦ . While this is a non-Euclidean geometry, spherical geometry develops along a separate path where
the axioms and theorems of neutral geometry do not typically apply.
The axioms and theorems of netural geometry apply to Euclidean and hyperbolic geometries. The
theorems below can be proven using the SMSG axioms 1 through 15. In the SMSG axiom list, Axiom
16 is the Euclidean parallel postulate. A neutral geometry assumes only the first 15 axioms of the
SMSG set.
Notes on notation: The SMSG axioms refer to the length or measure of line segments and the measure
of angles. Thus, we will use the notation AB to describe a line segment and AB to denote its length
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or measure. We refer to the angle formed by AB and AC as ∠BAC (with vertex A) and denote its
measure as m∠BAC.
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Lines and Angles
Definitions: Congruence
• Segments and Angles. Two segments (or angles) are congruent if and only if their measures
are equal.
• Polygons. Two polygons are congruent if and only if there exists a one-to-one correspondence
between their vertices such that all their corresponding sides (line sgements) and all their corresponding angles are congruent.
Theorem 1 Congruence relations are equivalance relations. Specifically, they are symmetric, reflexive
and transitive. †
Definitions: Points and Lines
• Collinear Two points are collinear if they lie on the same line.
• Intersecting Lines. Two lines intersect if there exists a point that is on both lines.
• Parallel Lines. Parallel lines are two lines in the same plan which do not intersect.
• Concurrent Lines. Concurrent lines are three or more coplanar lines that have a point in
common.
• Between. A point B is between A and C if B ∈ AC but B 6= A and B 6= C.
• Convex. A set (or polygon) P is convex if A, B ∈ P =⇒ AB ∈ P .
• Segment. A segment AB is the set of points A, B and all of the points between A and B.
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• Ray.A ray AB (also called a half-line) is a subset of the line AB that contains a given point
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A and all the points C ∈ AB such that A is not between C and B. The point A is called the
endpoint of the ray.
• Angle. An angle is the union of two noncollinear rays which have the same endpoint.
• Midpoint. The midpoint C of AB lies on AB such that AC = CB.
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• Angle Bisector. The the ray BD is the angle bisector of ∠ABC if m∠ABD = m∠DBC. Note
that angles must have measure ≤ 180◦ , so reflex angles are ruled out of consideration.
Theorem 2 Every line segment has exactly one midpoint. Every angle has exactly one bisector. †
Theorem 3 Supplements and complements of the same angles are congruent. †
Theorem 4 Vertical angles are congruent. †
Theorem 5 Pasch’s Axiom. Given a line that contains no vertex of a triangle, if that line intersects
one side of the triangle, it must intersect another. †
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Theorem 6 Crossbar Theorem. If AD is between AC and AB, then AD intersects BC.
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Triangles
Triangles exist in a neutral geometry and have many familiar properties from Euclidean geometry. The
lack of a parallel postuatle results in triangles whose angle sum is less than or equal to 180◦ . The
fact that many common notions about triangles are true in netural goemetries helps explain why the
ancients misunderstood Euclid’s parallel postulate and its consequences.
Theorem 7 Isosceles Triangle Theorem. If two sides of a triangle are congruent, then the angles
opposite these sides are congruent.†
Theorem 8 A point is on the perpendicular bisector of a line segment if and only if it is equidistant
from the endpoints of the line segment.
Definitions: Triangles
• Triangle. A triangle is the union of three line segments determined by three non-collinear points.
• Exterior Angle. Any angle that is both supplementary and adjacent to an angle of a triangle
(or polygon) is an exterior angle of the triangle (or polygon).
• Scalene Triangle. The sides of a scalene triangle all have different lengths.
Theorem 9 Each exterior angle of a triangle is greater in measure than either of the nonadjacent
interior angles of the triangle.
Theorem 10 ASA Triangle Congruence. If two angles and the included side of one triangle are
congruent, repsectively, to two angles and the included side of a second triangle, then the triangles
are congruent.
Theorem 11 Converse of Isosceles Triangle Theorem. If two angles of a triangle are congruent,
then the sides opposite these angles are congruent.
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Theorem 12 AAS Triangle Congruence. If the vertices of two triangles are in one-to-one correspondence such that two angles and the side opposite in one triangle are congruent to their
corresponding parts of a second triangle, then the triangles are congruent.
Theorem 13 SASAS Quadrilateral Congruence. If the vertices of two convex quadrilaterals
are in one-to-one correspondence such that three sides and the two included interior angles of
one quadrilateral are congrent to the corresponding parts of a second quadrilateral, then the
quadrilaterals are congruent.
Theorem 14 Inverse of Isosceles Triangle Theorem. If two sides of a triangle are not congruent,
then angles opposite those sides are not congurent, and the larger angle is opposite the larger
side.
Corollary The largest (smallest) angle of a triangle is opposite the largest (smallest) side.
Theorem 15 Triangle Inequality. The sum of the measures of two sides of a triangle is greater
than the measure of the third side.
Theorem 16 Hinge Theorem. If two sides of one triangle are congruent to two sides of another
triangle, and the included angle of the first is larger than the included angle of the second, then
the third side of the first triangle is longer than the third side of the second triangle. (Note that
this is Euclid’s Proposition 24 and is sometimes called the “Open Mouth Theorem.”)
Theorem 17 SSS Triangle Congruence. If the sides of one triangle are congruent to corresponding
sides of another triangle, the triangles are congruent.
Theorem 18 The angles of a scalene triangle all have different measures, and the angle with the
largest measure is opposite the side with the largest length.
Theorem 19 Prove or disprove: The analog of the SSS congruence condition for triangles is true for
quadrilaterals (SSSS).
Theorem 20 Two right triangles are congruent if the hypotenuse and one leg of one triangle are
congruent to corresponding sides of the other.
Theorem 21 Equilateral trianges are equiangular.
Theorem 22 Alternate Interior Angle Theorem. If two lines are intersected by a transversal
such that a pair of alternate interior angles is congreuent, then the lines are parallel.
Corollary 22.1 Two lines perpendicular to the same line are parallel.
Corollary 22.2 If two lines are intersected by a transversal such that a pair of corresponding
angles is congreuent, then the lines are parallel.
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Results of Euclid’s Parallel Postulate
Euclid’s Fifth Postulate says: “That, if a straight line falling on two straight lines makes the interior
angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet
on that side on which are the angles less than the two right angles.” Awkward, and it doesn’t mention
a parallel condition at all. Modern texts use the Playfair Postulate which is equivalent.
The Scottish Mathematician John Playfair reformulated Eculid’s Fifth Postulate in the late 1700’s, and
the SMSG axiom set (Axiom 16) uses a version of the Playfair Postulate: “Through a given external
point there is at most one line parallel to a given line.” We will use SMSG Axiom as Playfair’s Postulate
and assume the following theorem is true:
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Theorem 23 Euclid’s parallel postulate is equivalent to Playfair’s Postulate and SMSG Axiom 16.
Theorem 24 The Playfair Postulate (and Euclid’s parallel postulate) is equivalent to the Converse of
the Alternate Interior Angle theorem (Theorem 23).
Theorem 25 Euclid’s parallel postulate is equivalent to the statement: If a line intersects one of two
parallel lines, then it intersects the other.
Theorem 26 Euclid’s parallel postulate is equivalent to the statement: If a line is perpendicular to
one of two parallel lines, then it is perpedicular to the other.
Theorem 27 Wallis’s Postulate. Euclid’s parallel postulate is equivalent to the statement: Given
a 4ABC and any line segment DE, there exists a triangle having DE as one of its sides that is
similar to 4ABC.
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The Saccheri-Legendre Theorem
Theorem 28 The angle sum of any triangle is less than or equal to 180◦ .
Lemma 28.1 The sum of the measures of any two angles of a triangle is less than 180◦ .
Lemma 28.2 For any 4ABC there exists 4A1 B1 C1 having the same angle sum as 4ABC but
where m∠A1 = 21 m∠A.
Corollary 28.3 The angle sum of any convex quadrilateral is less than or equal to 360◦ .
An important result of the angle sum being no more than 180◦ is the following theorem:
Theorem 29: Unique Perpendicular Given a line l and any point P not on l, there exists a unique
line m such that P ∈ m and m ⊥ l.
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Rectangles
Offering the standard definitions of polygons and quadrilaterals is getting a bit ahead of ourselves since
most of figures defined below don’t necessarily exist in a neutral geometry. As soon as we have a
parallel postulate, however, they make sense.
Definitions: Quadrilaterals
• Polygon. A polygon is a simple, closed curve comprised entirely of line segments.
• Convex. For any two points in the interior of a convex polygon, the line segment connecting
those points lies wholly within the interior of the polygon.
• Quadrilateral. A quadrilateral is a polygon containing exactly four line segments.
• Trapezoid. A trapezoid is a convex quadrilateral with exactly one pair of parallel sides.
• Parallelogram. A parallelogram is a convex quadrilateral with two pairs of parallel sides.
• Rectangle. A rectangle is a parallelogram with a right angle.
• Kite. A kite is a convex quadrilateral with two pairs of adjacent, congruent sides.
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• Rhombus. A rhombus is a parallelogram with opposite sides congruent.
5.1
Searching for Rectangles
Rectangles don’t neccessarily exist in a neutral geometry. In fact, the existence of a rectangle is
equivalent to the Euclidean parallel postulate. Two pioneers tried to use the idea of a rectangle to
prove Euclid’s Fifth Postulate: Saccheri and Lambert.
Definitions: Saccheri and Lamber Quadrilaterals
• Saccheri Quadrilateral. A Saccheri quadrilateral is a quadrilateral with two congruent sides
which are both perpendicular to a third side, called the base.
• Lambert Quadrilateral. A Lambert Quadrilateral is a quadrilateral with at least three right
angles.
Both Saccheri and Lambert formulated their quadrilaterals as an attempt to prove the Euclidean
Parallel Postulate was a theorem. Of course, it’s not, and they failed. However, their investigations led
to deepened understanding of the implications of a parallel postulate and the creation of non-Euclidean
geometries.
Theorem 30 The diagonals of a Saccheri quadrilateral are congruent.
Theorem 31 The summit angle of a Saccheri quadrilateral are congruent.
Theorem 32 The summit angle of a Saccheri quadrilateral are not obtuse and thus are either right
or acute.
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Theorem 33 The lines joining the midpoints of the summit and base of a Saccheri quadrilateral are
perpendicular to both.
Corollary 32.1 The summit and base of a Saccheri quadrilateral are parallel.
Theorem 34 In any Saccheri quadrilateral, the length of the summit is greater than or equal the
length of the base.
Theorem 35 The fourth angle of a Lambert quadrilateral is not obtuse, and is therefore either right
or acute.
Theorem 36 The measure of the side included between two right angles of a Lambert quadrilateral
is less than or equal to the measure of the side opposite it.
Both Lambert and Saccheri were successful in eliminating the possibility of the fourth side of a “rectangle” being an obtuse angle, but neither could eliminate the possibility that it was acute. Their work
led to our understanding that the existence of a rectangle was, in fact, equivalent to the Euclidean
Parallel Postulate.
5.2
Rectangles and the Euclidean Parallel Postulate
Theorem 37 If one rectangle exists, then there exists a rectangle with an arbitrarily large side.
Corollary 36.1 If one rectangle exists, then there exists a rectangle with two arbitrarily large
sides.
Theorem 38 If one rectangle exists, then there exists a rectangle with adjacent sides congruent to
the given line segments AB and DC.
Theorem 39 If one rectangle exists, then every triangle has an angle sum of 180◦ .
Theorem 40 If one triangle having angle sum 180◦ exists, then a rectangle exists.
Corollary 40.1 If one triangle having angle sum 180◦ exists, then all triangles have an angle
sum of 180◦ .
Due to the previous theorems in this section, we can prove two more statements are equivalent to the
Euclidean parallel postulate.
Theorem 41 Euclid’s parallel postulate is equivalent to the statement: The angle sum of every triangle
is 180◦ .
Theorem 42 Euclid’s parallel postulate is equivalent to the Pythagorean Theorem.
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