Triangle postulate
Encyclopedia
In Euclidean geometry
, the triangle postulate states that the sum of the angles of a triangle
is two right angle
s. This postulate is equivalent to the parallel postulate
. The following statements are equivalent:
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
, the triangle postulate states that the sum of the angles of a triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
is two right angle
Right angle
In geometry and trigonometry, a right angle is an angle that bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles...
s. This postulate is equivalent to the parallel postulate
Parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...
. The following statements are equivalent:
- Triangle postulate: The sum of the angles of a triangle is two right angles.
- Playfair's axiomPlayfair's axiomPlayfair's axiom is a geometrical axiom, intended to replace the fifth postulate of Euclides :Given a line and a point not on it, at most one parallel to the given line can be drawn through the point....
: 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.