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Definition
| any two lines meet and every line has at least three distinct points lying on it |
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| add enough new "points at infinity" so that all lines parallel to any given line will now meet at one such point |
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| 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 also intersects either AC or BC |
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| an isosceles bi-right quadrilateral ABDC (one whose sides are congruent CA is congruent to DB) |
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| a quadrilateral with at least 3 right angles |
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| The Euclidean plane with one ideal point appended to it |
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| cut each other at right angles |
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| Two triangles are congruent if their SAS are congruent. Euclid was not happy with this, however, so he used superposition (placing a triangle on top of another triangle) to see if they are congruent. |
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| A model of our incidence, betweenness, and congruence axioms is called a Hilbert Plane |
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| Alternate Interior Angle (AIA) Theorem |
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Definition
| In any Hilbert plane, if two lines cut by a transversal have a pair of congruent alternate interior angles with respect to that transversal, then the two lines are parallel |
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Definition
| If ray AD is between ray AC and ray AB, then ray AD intersects segment BC |
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| The "No Similarity" Theorem |
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Definition
| In a plane satisfying the acute angle hypothesis, if two triangles are similar, then they are congruent. In other words, AAA is a valid criterion for congruence of triangles. |
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| Circle-Circle Continuity Principle |
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Definition
| If a circle gamma has one point inside and one point outside another circle gamma', then the two circles interest in two points. |
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| Line-Circle Continuity Principle |
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Definition
| If a line passes through a point inside a circle, then the line intersects the circle in two points |
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Definition
| If all points of a straight line fall into two classes, such that every point of the first class lies to the left of any point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions |
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Definition
| Given a line l and a point P on l. Let Q be the foot of the perpendicular from P to l. A limiting parallel ray to l eminating from P is a ray PX that does not intersect l and such that for every ray PY which is between ray PQ and ray PX, ray PY intersects l. |
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Definition
| If two angles add up to 90 degrees |
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| Hilbert's Euclidean Parallel Postulate |
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Definition
| The degree of an exterior angle to a triangle is equal to the sum of the degrees of its remote interior angles. |
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Definition
| For any Hilbert plane, if one Saccheri quadrilateral has acute (or right, obtuse) summit angles, then so do all Saccheri quadrilaterals |
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Definition
| A Hilbert plane is semi-Euclidean if all Lambert quads and all Saccheri quads are rectangles. |
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| Saccheri-Legendre Theorem |
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Definition
| In an Archimedean Hilbert plane, the angle sum of every triangle is less than or equal to 180 degrees |
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