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In Euclidean geometry, the triangle postulate states that the sum of the angles of a triangle is two right angles. This postulate is equivalent to the parallel postulate. The following statements are equivalent:
- Triangle postulate: The sum of the angles of a triangle is two right angles.
- Playfair's axiom: Given a straight line and a point not on the line, exactly one straight line may be drawn through the point parallel to the given line.
- Proclus' axiom: If a line intersects one of two parallel lines, it must intersect the other also.
- Equidistance postulate: Parallel lines are everywhere equidistant.
- Triangle area property: The area of a triangle can be as large as we please.
- Three points property: Three points either lie on a line or lie on a triangle.
- Pythagoras' theorem: In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
- Eric W. Weisstein (2003). CRC concise encyclopedia of mathematics (2nd ed.). p. 2147. ISBN 1-58488-347-2. "The parallel postulate is equivalent to the Equidistance postulate, Playfair axiom, Proclus axiom, the Triangle postulate and the Pythagorean theorem."
- Keith J. Devlin (2000). The Language of Mathematics: Making the Invisible Visible. Macmillan. p. 161. ISBN 0-8050-7254-3.
- Alexander R. Pruss (2006). The principle of sufficient reason: a reassessment. Cambridge University Press. p. 11. ISBN 0-521-85959-X. "We could include...the parallel postulate and derive the Pythagorean theorem. Or we could instead make the Pythagorean theorem among the other axioms and derive the parallel postulate."