I'm looking for a basic proof of the basic plane Euclidean geometry theorem: Two triangles are similar if and only if their corresponding (interior) angles are equal.

(The theorem can be also worded in terms of only two of the interior angles.)

I'm having great difficulty finding an "elementary" proof of this theorem. (I have searched both online and in my local university library.)

By "elementary" I mean a proof that uses only "the standard axioms" of plane Euclidean geometry (as taught in basic high-school geometry). (In particular, I'm specifically ruling out proofs that are based on analytic geometry.)

I have been able to find some proofs online, but they all depend on the theorem that "two transversals are divided into proportional segments by a system of parallel lines." Unfortunately, every proof that I can find of this particular theorem happens to be based on the subject line's theorem.

Can someone point me to a reference where this theorem is proved in full?


When I write "two triangles $ABC$ and $DEF$ are similar" I mean that the ratios of the lengths of their (corresponding) sides are equal, i.e.,

$$\overline{AB}:\overline{BC}:\overline{CA} = \overline{DE}:\overline{EF}:\overline{FD}$$

  • 5
    $\begingroup$ What is your definition for two triangles to be similar? If it is not that they have corresponding angles, I take it that is that the triangles have sidelengths in the same ratio? $\endgroup$ – JavaMan Jun 16 '13 at 17:09
  • $\begingroup$ (@JavaMan: see my EDIT.) $\endgroup$ – kjo Jun 16 '13 at 18:04
  • $\begingroup$ What is your definition of two angles are equal ? In W. Schwabhäuser W Szmielew A. Tarski Metamathematische Methoden in der Geometrie by definition two angles ABC DEF are congruent if: exists A', exists C', exists D', exists F', Bet B A A' /\ Cong A A' E D /\ Bet B C C' /\ Cong C C' E F /\ Bet E D D' /\ Cong D D' B A /\ Bet E F F' /\ Cong F F' B C /\ Cong A' C' D' F'. With this definition there is almost nothing to prove. $\endgroup$ – Julien Narboux Jun 22 '15 at 8:20

How about Euclid book 6? One direction is Proposition 4. The other direction is Proposition 5.

If Euclid's rigor is not good enough for you, I'm sure this can be found in Hilbert's Grundlagen also, but I don't have a copy here to give you the exact reference.


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