PROPOSITIONS

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 half-planes have no point in common.
Lemma 1	Given ABC 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 ABC and ACD. Then BCD and ABD.
Corollary	Given ABC and BCD. Then ABD and ACD.
Proposition 4	If BAC 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 or on the opposite ray .
Pasch's Thm.	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.
Proposition 5	Given ABC. Then AC=ABBC and B is the only point common to segments AB and BC.
Proposition 6	Given ABC. Then B is the only point common to rays and , and .
Proposition 7	Given angle and a point D lying on line . Then D is in the interior of angle if and only if BDC.
Proposition 8	If D is in the interior of angle then: so is every other point on ray except A. no point on the opposite ray to is in the interior of angle . if CAE, then B is in the interior of angle .
Crossbar Theorem	If is between and , then intersects segment BC.
Proposition 9	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. 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 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 , there is a unique point F on a given side of line such that .
Proposition 9	(Segment Subtraction) If ABC, DEF, , and then .
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, , or AB>CD. (b) If AB<CD and , then AB<EF. (c) If AB>CD and , then AB>EF. (d) If AB<CD and CD<EF, then AB<EF.
Proposition 12	(Angle Addition) Given between and , between and , , and . Then .
Proposition 13	(Angle Subtraction) Given between and , between and , , and . Then .
Proposition 14	(Ordering of Angles) (a) ) Exactly one of the following conditions holds (trichotomy): . (b) . (c) . (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.
Corollary 1	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.
Corollary 2	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.
Theorem 2	(Exterior Angle Theorem) An exterior angle of a triangle is greater than either remote interior angle.
Proposition 1	(SAA) If , , and , then .
Proposition 2	(HL) Two right triangles are congruent if the hypotenuse and a leg are congruent respectively to the hypotenuse and a leg of the other.
Proposition 3	Every segment has a unique midpoint.
Proposition 4	Every angle has a unique bisector.
Proposition 5	Every segment has a unique perpendicular bisector.
Proposition 6	In every triangle the greater angle lies opposite to the greater side, and the greater side lies opposite to the greater angle.
Proposition 7	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.
Theorem 3	A. There is a unique way of assigning degree measures to each angle such that the following properties hold: The measure of each angle is a real number between o and 180⁰. The measure of an angle is 90⁰ if and only if the angle is a right angle. Two angles have the same measure if and only if they are congruent to each other. If is interior to angle , then For every real number x between 0 and 180, there exists an angle whose measure is 180⁰. If two angles are supplementary, then the sum of their measures is 180⁰. 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: The length of each segment is a positive real number, and the length of the unit segment is 1. Two segments have the same length if and only if they are congruent to each other. ABC if and only if . 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. For every positive real number x, there is a segment of length x.
Corollary 1	The sum of the degree measures of two angles of a triangle is less than 180⁰.
Corollary 2	(Triangle Inequality) If A, B, and C are three non-collinear points, then: .
Theorem 4	(Saccheri-Legendre) The sum of the degree measures of the three angles in any triangle is less than or equal to 180⁰.
Corollary 1	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.
Corollary 2	The sum of the degree measures of the angles in any convex quadrilateral is at most 360⁰.
Theorem 5	The following statements are equivalent: Euclid's Fifth Postulate. Hilbert's Parallel Postulate. If a line intersects one of two parallel lines, then it also intersects the other. The converse of Alternate Interior Theorem (theorem 1). If t is transversal to l and m, l and m are parallel, t and l are perpendicular, then t and m are perpendicular. 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. The angle sum of every triangle is 180⁰.
Theorem 6	(Additivity of the Defect) If A, B, and C are vertices of a triangle, and D is a point between A and B, then: .
Corollary	If A, B, and C are vertices of a triangle, and D is a point between A and B, then the angle sum of is 180⁰ if and only if the angle sum of both and are equal to 180⁰.
Theorem 7	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.
Corollary	If there exists a triangle with positive defect, then all triangles have positive defect.

HYPERBOLIC GEOMETRY

Lemma 1	Rectangles do not exist.
Theorem 1	(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.
Corollary	For every line l and every point P not on l, there are infinitely many parallels to l through P.
Theorem 2	The sum of the degree measures of the three angles in any triangle is less than 180⁰.
Corollary	The sum of the degree measures of the angles in any convex quadrilateral is less than 360⁰.
Theorem 3	(AAA) If the three angles of a triangle are congruent respectively to the three angles of another triangle, then the two triangles are congruent.