Legendre’s theorem on angles of triangle

Adrien-Marie Legendre has proved some theorems concerning the sum of the angles of triangle.  Here we give one of them, being the inverse of the theorem in the entry “sum of angles of triangle in Euclidean geometryMathworldPlanetmath”.

Theorem.  If the sum of the interior anglesMathworldPlanetmath of every triangle equals straight angleMathworldPlanetmath, then the parallel postulate is true, i.e., in the plane determined by a line and a point outwards it there is exactly one line through the point which does not intersect the line.

Proof.  We consider a line a and a point B not belonging to a.  Let BA be the normal line of a (with Aa) and b be the normal line of BA through the point B.  By the supposition of the theorem, b does not intersect a.

We will show that in the plane determined by the line a and the point B, there are through B no other lines than b not intersecting the line a.  For this purpose, we choose through B a line b which differs from b; let the line b form with BA an acute angleMathworldPlanetmath β.

We determine on the line a a point A1 such that  AA1=AB.  By the supposition of the theorem, in the isosceles right triangle BAA1 we have


Next we determine on a a second point A2 such that  A1A2=A1B.  By the supposition of the theorem, in the isosceles triangleMathworldPlanetmath BA1A2 we have


We continue similarly by forming isosceles triangles using the points A3, A3, , An of the line a such that


Then the acute angles being formed beside the points are


They form a geometric sequence with the common ratior=12.  When n is sufficiently great, the member αn is less than any given positive angle.  As we have so much triangles BAn-1An that  αn<π2-β,  then


Then the line b falls after penetrating B into the inner territory of the triangle ABAn.  Thereafter it must leave from there and thus intersect the side AAn of this triangle.  Accordingly, b intersects the line a.

The above reasoning is possible for each line  bb  through B.  Consequently, the parallel axiom is in force.


  • 1 Karl Ariva: Lobatsevski geomeetria.  Kirjastus “Valgus”, Tallinn (1992).
Title Legendre’s theorem on angles of triangle
Canonical name LegendresTheoremOnAnglesOfTriangle
Date of creation 2013-05-11 13:55:53
Last modified on 2013-05-11 13:55:53
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Theorem
Classification msc 51M05