List of Theorems Using Moise's Axioms

Theorem (1) If and belong to line and then

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

Theorem (4) If and hold, then is true.

Theorem (5) If and lie on line then if and only if or

Theorem (6) If lies on ray and then

Theorem (7) If there exists a unique point on ray such that and

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

Theorem (13) If there is a unique ray such that and

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

Theorem (17) If then there exists a unique perpendicular to line at

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

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

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

Theorem (27) If lies in the interior of then ray meets segment at some interior point

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

Theorem (30) For any two angles and there is a unique ray on the of line such that

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

Theorem (34) If and then

Theorem (35) In if and only if

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

Theorem (38) If and is the midpoint of segment then the line is perpendicular to the segment

Theorem (39) If and is the midpoint of segment then the ray bisects

Theorem (40) The set of all points equidistant from two distinct points and is the perpendicular bisector of the segment

Theorem (41) If and then

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

Theorem (49) Given with equality only if

Theorem (50) If is the midpoint of

Theorem (51) If in and then if and only if

Theorem (52) If and then

Theorem (53) If and then and are supplementary angles.

Theorem (54) In acute angles triangles, if and then

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