Exploration of non-Euclidean Geometries
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Propositions
Below are modern translations of the first 32 of Euclid's postulates:
- From any line segment an equilateral triangle can be created.
- Given a line segment, an additional equal line segment can be drawn at the first segment’s end point.
- Given two unequal line segments it is possible to cut off a portion of the larger one to make the segments equal.
- If two triangles have two respective sides equal and their contained angles equal, then two triangles are congruent.
- An isosceles triangle has equal base angles. Additionally, when the side lengths are extended, the exterior base angles will be equal.
- In a triangle containing two equal angles the sides opposing the angles will also be equal.
- Given a triangle, it is impossible to construct a second triangle with the same base and respective side lengths on the same side of the base whose sides meet at a different point than the original triangle’s sides meet.
- If all three respected sides of two triangles are equal than all respected contained angles are similarly equal.
- It is possible to bisect an angle.
- It is possible to bisect a line segment.
- A straight line perpendicular to an original line can be drawn at any point on the original line.
- It is possible to draw a line perpendicular to an original line through any point not on the original line.
- Two intersecting straight lines are either perpendicular or have a same side angle sum equal to 180.
- If two straight lines on opposing sides of a different straight line have adjacent angles that sum to 180, then the first two lines are a single straight line.
- Vertical angles created by two intersecting lines are equal.
- In a triangle with an extended side, the exterior angle created is larger than the two opposite interior angles.
- In any triangle, the sum of any two angles will be less than 180
- In a triangle, the angle opposite the longest side is the biggest.
- In a triangle, the side opposite the biggest angle is longest.
- The sum of the lengths of any two sides of a triangle is greater than the length of the remaining side.
- Given a triangle ABC, construct two lines originating at the vertices B and C, intersecting at point D which exists within triangle ABC as shown in Figure 1. The sums of the related segments and are less than the sum of the original segments and , although the measure of BDC will be greater than that of BAC.
- Given three line segments in which the sum of any two segments is greater than the remaining segment, a triangle can be constructed with segments of equal length to the given segments.
- Given an angle, another equal angle can be created at a point on a given line.
- If two triangles have two related sides with the same lengths, then the triangle in which these sides contain the larger angle will have a larger base.
- Given two triangles with two related sides of the same length, then the triangle in which the base is larger will have a larger angle contained in those sides.
- In two triangles, if two given related angles are equal and one relating side is also equal than the other related sides and angle are equal causing the two triangles to be congruent.
- If a straight line intersects two additional straight lines, and the alternate angles created are equal to one another, then the two additional straight lines are parallel.
- If a straight line intersects two additional straight lines, and the exterior angle is equal to the opposite interior angle on the same side, or if the sum of two interior angles on the same side are equal to 180, then the two additional straight lines are parallel to one another.
- A straight line intersecting two parallel straight lines causes the alternate angles to be equal to one another, the exterior angle equal to the interior opposite angle, and sum of the interior angles on the same side to be equal to 180
- Straight lines that are parallel to a different line are also parallel to each other.
- Given a straight line and a point not on the line it is possible to create a line parallel to the first through the given point.
- If one side of a triangle is extended, then the exterior angle is equal to the sum of the opposite interior angles. In addition, all three interior angles sum to 180