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PROPOSITIONS
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PROPOSITIONS
INCIDENCE
Proposition 1
If l and m are two distinct lines that are not parallel, then l and m have
a unique point in common.
Proposition 2
There exist three distinct lines that are not concurent.
Proposition 3
For every line there is at least one point not lying on it.
Proposition 4
For every point there is at least one line not passing through it.
Proposition 5
For every point P there exist at least two lines through P.
BETWEENNESS
Proposition 1
For any two points A and B:
(i)
(ii)
Proposition 2
Every line bounds exactly two half-planes, and these halfplanes have no point in common.
Lemma 1
Given A*B*C and l a line through A that does not contain B
and C, then B and C are on the same side of l.
Proposition 3
Given A*B*C and A*C*D. Then B*C*D and A*B*D.
Corollary
Given A*B*C and B*C*D. Then A*B*D and A*C*D.
Proposition 4
If B*A*C and l is a line through A, B, and C (the existence
of l is guaranteed by B-1), then for any point P lying on l, P
lies either on ray
Pasch's Thm.
or on the opposite ray
.
If A, B, C are distinct non-collinear points and l is any line
intersecting AB in a point between A and B, then l intersects
either AC or BC. If C does not lie on l, then l does not
intersect both AC and BC.
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Proposition 5
Proposition 6
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Given A*B*C. Then AC=AB BC and B is the only point
common to segments AB and BC.
Given A*B*C. Then B is the only point common to rays
and
Proposition 7
, and
.
Given angle
and a point D lying on line
D is in the interior of angle
Proposition 8
if and only if B*D*C.
If D is in the interior of angle
then:
a. so is every other point on ray
b. no point on the opposite ray to
angle
. Then
except A.
is in the interior of
.
c. if C*A*E, then B is in the interior of angle
Crossbar Theorem
Proposition 9
If
BC.
is between
and
, then
.
intersects segment
a. If a ray emanating from an exterior point of triangle
intersects side AB in a point between A and B,
then it also intersects side AC or side BC.
b. If a ray emanates from an interior point of triangle
, then it intersects one of the sides, and if it
does not pass through a vertex, it intersects only one
side.
CONGRUENCE
Proposition 1
The base angles of an isosceles triangle are congruent.
Proposition 2
Supplements of congruent angles are congruent.
Proposition 3
Vertical angles are congruent to each other.
Proposition 4
An angle congruent to a right angle is a right angle.
Proposition 5
For every line l and point P there exist a line through P perpendicular
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to l.
Proposition 6
(ASA) If two angles and the included side of a triangle are congruent
respectively to two angles and the included side of another triangle,
then the two triangles are congruent.
Proposition 7
If two angles in triangle are congruent, then the triangle is isosceles.
Proposition 8
(Corollary to SAS) Given
and segment
unique point F on a given side of line
, there is a
such that
.
Proposition 9
(Segment Subtraction) If A*B*C, D*E*F,
then
, and
.
Proposition 10
Given
then for every point B between A and C, there exist
a point E between D and F such that
.
Proposition 11
(Segment Ordering) (a) Exactly one of the following conditions holds
(trichotomy): AB<CD,
Proposition 12
, or AB>CD. (b) If AB<CD and
, then AB<EF. (c) If AB>CD and
AB>EF. (d) If AB<CD and CD<EF, then AB<EF.
, then
(Angle Addition) Given
between
and
,
between
and
,
, and
. Then
.
Proposition 13
(Angle Subtraction) Given
and
,
between
and
, and
,
between
. Then
.
Proposition 14
(Ordering of Angles) (a) ) Exactly one of the following conditions
holds (trichotomy):
.
(b)
.
(c)
.
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(d)
.
Proposition 15
(SSS) If the three sides of a triangle are congruent respectively to the
three sides of another triangle, then the two triangles are congruent.
Proposition 16
(Euclid's Fourth Postulate) All right angles are congruent to each
other.
NEUTRAL GEOMETRY
Theorem 1
(Alternate Interior Angle Theorem) If two lines cut by a
transversal have a pair of congruent alternate interior
angles, then the two lines are parallel.
Two distinct lines perpendicular to the same line are
parallel. The perpendicular dropped from a point P not on
a line l to l is unique.
For every line l and point P not lying on l there is at least
one line m through P such that m is parallel to l.
(Exterior Angle Theorem) An exterior angle of a triangle
is greater than either remote interior angle.
Corollary 1
Corollary 2
Theorem 2
Proposition 1
Proposition 2
Proposition
Proposition
Proposition
Proposition
3
4
5
6
Proposition 7
Theorem 3
(SAA) If
,
, and
, then
.
(HL) Two right triangles are congruent if the hypotenuse
and a leg are congruent respectively to the hypotenuse
and a leg of the other.
Every segment has a unique midpoint.
Every angle has a unique bisector.
Every segment has a unique perpendicular bisector.
In every triangle the greater angle lies opposite to the
greater side, and the greater side lies opposite to the
greater angle.
If two sides of a triangle are congruent respectively to two
sides of another triangle, then the included angle in the
first triangle is less than the included angle in the second
triangle if and only if the third side in the first triangle is
less than the third side in the second triangle.
A. There is a unique way of assigning degree measures to
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each angle such that the following properties hold:
0. The measure of each angle is a real number between
o and 1800.
1. The measure of an angle is 900 if and only if the
angle is a right angle.
2. Two angles have the same measure if and only if
they are congruent to each other.
3. If
is interior to angle
, then
4. For every real number x between 0 and 180, there
exists an angle whose measure is 1800.
5. If two angles are supplementary, then the sum of
their measures is 1800.
6. An angle is larger than another angle if and only if
the measure of the first angle is larger than the
measure of the second angle.
B. Given a segment OI called the unit segment,
there is a unique way of assigning a length
to
each segment AB such that the following properties
hold:
7. The length of each segment is a positive real
number, and the length of the unit segment is 1.
8. Two segments have the same length if and only if
they are congruent to each other.
9. A*B*C if and only if
.
10. The length of one segment is less than the length of
another segment if and only if the first segment is
less than the second segment.
11. For every positive real number x, there is a segment
of length x.
Corollary 1
Corollary 2
Theorem 4
The sum of the degree measures of two angles of a
triangle is less than 1800.
(Triangle Inequality) If A, B, and C are three noncollinear points, then:
.
(Saccheri-Legendre) The sum of the degree measures of
the three angles in any triangle is less than or equal to
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Corollary 1
Corollary 2
Theorem 5
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1800.
The sum of the degree measures of two angles in a
triangle is less than or equal to the degree measure of
their remote exterior angle.
The sum of the degree measures of the angles in any
convex quadrilateral is at most 3600.
The following statements are equivalent:
1. Euclid's Fifth Postulate.
2. Hilbert's Parallel Postulate.
3. If a line intersects one of two parallel lines, then it
also intersects the other.
4. The converse of Alternate Interior Theorem
(theorem 1).
5. If t is transversal to l and m, l and m are parallel, t
and l are perpendicular, then t and m are
perpendicular.
6. If k and l are parallel, m and k are perpendicular,
and n and l are perpendicular, then either m=n, or m
and n are parallel.
7. The angle sum of every triangle is 1800.
Theorem 6
Corollary
(Additivity of the Defect) If A, B, and C are vertices of a
triangle, and D is a point between A and B, then:
.
If A, B, and C are vertices of a triangle, and D is a point
is 1800 if
between A and B, then the angle sum of
and only if the angle sum of both
equal to 1800.
Theorem 7
Corollary
and
are
If a triangle whose angle sum is 1800, then a rectangle
exists. If a rectangle exists, then every triangle has angle
sum equal to 1800.
If there exists a triangle with positive defect, then all
triangles have positive defect.
HYPERBOLIC GEOMETRY
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Lemma 1
Theorem 1
Corollary
Theorem 2
Corollary
Theorem 3
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Rectangles do not exist.
(Universal Hyperbolic Theorem) For every line l and
every point P not on l, there exist at least two distinct
parallel lines to l that pass through P.
For every line l and every point P not on l, there are
infinitely many parallels to l through P.
The sum of the degree measures of the three angles in
any triangle is less than 1800.
The sum of the degree measures of the angles in any
convex quadrilateral is less than 3600.
(AAA) If the three angles of a triangle are congruent
respectively to the three angles of another triangle, then
the two triangles are congruent.
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