Angles in Triangle Add to 180°

Based on the Parallel postulate, Euclid proves in Proposition I.32 - the basic fact of Euclidean geometry - namely, that the sum of the angles in a triangle is always equal to "two right angles". The proof is not difficult, although Sir Thomas L. Heath in his commentary gives an additional proof that is even simpler. The applet below illustrates a proof by "paper folding".


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Let A denote the vertex with the largest (or one of the largest) angles. Then the feet D of the altitude AD is located on the base BC, between B and C. D is constructed via Axiom O4. Folds BC, EG and FH exist due to Axiom O2.

Now, why the construction works? AD ⊥ BC and EF ⊥ AD, we see that EF||BC and that AD is divided by EF into equal halves. Therefore, EF is a midline of DABC:

(1)EF||BC
(2)BE = AE, CF = AF.

By construction, also, AE = DE and BE = DE. Therefore, the edges DE of the two foldings at EF and EG coincide. Similarly for DF and the foldings EF and FH. Thus the three angles at D, on one hand, add to two right angles and, on the other, are equal to the angles of DABC.

Paper Folding Geometry

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