Equivalences to Euclidu00b9s Fifth Postulate.docx -...

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Equivalences to Euclid’s Fifth Postulate (some of them)a.There is at most one line that can be drawn parallel to another given one through an external point. (Playfair’s axiom)b.The sum of the angles in every triangle is 180° (triangle postulate).c.There exists a triangle whose angles add up to 180°.d.The sum of the angles is the same for every triangle.e.There exists a pair of similar, but not congruent, triangles.f.Every triangle can be circumscribed.g.If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle.h.There exists a quadrilateral in which all angles are right angles. i.e. rectanglesexist.i.A line intersecting and perpendicular to one side of an acute angle intersects the other side.j.If angle ACB is a right angle, then Cis on the circle with diameterAB.k.There exists a point equidistant from any three noncollinear points.l.The perpendicular bisectors of the legs of a right triangle intersect.m.Every two parallel lines have a common perpendicular.n.The diagonals of a Saccheri quadrilateral bisect each other.o.If in a quadrilateral a pair of opposite sides are equal and if the angles adjacent to a third side are right angles, then the other two angles are also right angles.p.The circumference of any circle of radius r is 2r.πq.There exists a pair of straight lines that are at constant distance from each other.

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