List of Theorems Using Moise's Axioms
Theorem (1) If
and
belong to line
and
then
![list of theorems using moises axioms _gr_5.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_5.gif)
Theorem (2) Two distinct lines meet in at most one point; a line which meets a plane containing it intersects that plane in exactly one point.
Theorem (3) If
then
and neither
nor
![list of theorems using moises axioms _gr_9.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_9.gif)
Theorem (4) If
and
hold, then
is true.
Theorem (5) If
and
lie on line
then
if and only if
or
![list of theorems using moises axioms _gr_20.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_20.gif)
Theorem (6) If
lies on ray
and
then
![list of theorems using moises axioms _gr_24.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_24.gif)
Theorem (7) If
there exists a unique point
on ray
such that
and
![list of theorems using moises axioms _gr_29.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_29.gif)
Theorem (8) The midpoint of any segment exists, and is unique.
Theorem (9) If
then
Theorem (10) If
and
are three distinct points, collinear points, then either
or
Theorem (11) A segment cannot be ray.
Theorem (12) If the rays
and
have coordinates
and
relative to some half-plane, then
if and only if either
or
![list of theorems using moises axioms _gr_45.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_45.gif)
Theorem (13) If
there is a unique ray
such that
and
![list of theorems using moises axioms _gr_49.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_49.gif)
Theorem (14) The bisector of any angle exists and is unique.
Theorem (15) Angles supplementary (or complementary) to the same angles have the same measure.
Theorem (16) Two lines
and
are perpendicular at
if and only if
![list of theorems using moises axioms _gr_53.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_53.gif)
Theorem (17) If
then there exists a unique perpendicular to line
at
![list of theorems using moises axioms _gr_56.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_56.gif)
Theorem (18) Vertical angles have equal measures.
Theorem (19) Bisectors of a linear pair of angles are perpendicular.
Theorem (20) If
and
are any three rays on one side of a line and having the same end point, then either
or
![list of theorems using moises axioms _gr_61.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_61.gif)
Theorem (21) If two angles have a side in common that passes through an interior point of the angle formed by the other two sides, then the other two sides are perpendicular if and only if the given angles are complementary.
Theorem (22) If
holds and
passes through point
but not point
then
and
lie on opposite sides of line
![list of theorems using moises axioms _gr_68.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_68.gif)
Theorem (23) If point
lies on
and point
lies in one of the half planes determined by
then, except for
the entire segment
or ray
lies in that half-plane.
Theorem (24) Let
and
lie on opposite sides of a line
and let
and
be any two distinct points on
Then the segment
and ray
have no point in common.
Theorem (25) Suppose
and
are any three distinct noncollinear points in a plane, and
is any line in that plane that passes through an interior point
of one of the sides,
of the triangle determined by
and
Then line
meets either
at some interior point
the cases being mutually exclusive.
Theorem (26) If
and
lie on the sides of
then, except for the end points, segment
is a subset of the interior of
If
Interior
then, except for
ray
![list of theorems using moises axioms _gr_103.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_103.gif)
Theorem (27) If
lies in the interior of
then ray
meets segment
at some interior point
![list of theorems using moises axioms _gr_108.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_108.gif)
Theorem (28) Segments and rays are convex sets, but an angle is not.
Theorem (29) Suppose that
and
are distinct, noncollinear points and that
and
Prove that there exists a unique point
such that
and
![list of theorems using moises axioms _gr_115.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_115.gif)
Theorem (30) For any two angles
and
there is a unique ray
on the
of line
such that
![list of theorems using moises axioms _gr_121.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_121.gif)
Theorem (31) Every half-plane is a nonempty set.
Theorem (32) The congruence relations
for segments, angles and triangles are equivalence relations.
Theorem (33) If
and
then either
and
or
and
![list of theorems using moises axioms _gr_128.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_128.gif)
Theorem (34) If
and
then
![list of theorems using moises axioms _gr_132.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_132.gif)
Theorem (35) In
if and only if
![list of theorems using moises axioms _gr_135.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_135.gif)
Theorem (36) A triangle is isosceles if and only if base angles are congruent.
Theorem (37) If
is the midpoint of segment
and the line
is perpendicular to
then
![list of theorems using moises axioms _gr_140.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_140.gif)
Theorem (38) If
and
is the midpoint of segment
then the line
is perpendicular to the segment
![list of theorems using moises axioms _gr_145.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_145.gif)
Theorem (39) If
and
is the midpoint of segment
then the ray
bisects
![list of theorems using moises axioms _gr_150.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_150.gif)
Theorem (40) The set of all points equidistant from two distinct points
and
is the perpendicular bisector of the segment
![list of theorems using moises axioms _gr_153.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_153.gif)
Theorem (41) If
and
then
![list of theorems using moises axioms _gr_157.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_157.gif)
Theorem (42) Given
there exists a unique perpendicular from point
to line
Theorem (43) The measure of an exterior angle of a triangle is greater than that of either opposite interior angle.
Theorem (44) The sum of the measures of two angles of a triangle is less than 180.
Theorem (45) A triangle can have at most one right or obtuse angle.
Theorem (46) The base angles of an isosceles triangle are acute.
Theorem (47) The angle sum of a triangle is less than or equal to 180.
Theorem (48) Given
if and only if
![list of theorems using moises axioms _gr_163.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_163.gif)
Theorem (49) Given
with equality only if
![list of theorems using moises axioms _gr_166.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_166.gif)
Theorem (50) If
is the midpoint of
![list of theorems using moises axioms _gr_169.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_169.gif)
Theorem (51) If in
and
then
if and only if
![list of theorems using moises axioms _gr_175.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_175.gif)
Theorem (52) If
and
then
![list of theorems using moises axioms _gr_179.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_179.gif)
Theorem (53) If
and
then
and
are supplementary angles.
Theorem (54) In acute angles triangles, if
and
then
![list of theorems using moises axioms _gr_189.gif]](pages/list-of-theorems-using-moises-axioms/Images/list-of-theorems-using-moises-axioms_gr_189.gif)
Theorem (55) If the hypotenuse and leg of one right triangle are congruent to the hypotenuse of another, the triangles are congruent.
Theorem (56) If the hypotenuse and acute angle of one right triangle are congruent to the hypotenuse and acute angle of another, the triangles are congruent.
Theorem (57) If a leg and acute angle of ne triangle are congruent to the corresponding leg and acute angle of another, the triangles are congruent.
Theorem (58) Suppose that in
and
and
Then
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Cite this as: List Of Theorems Using Moises Axioms Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/list-of-theorems-using-moises-axioms.html
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