![]() ![]() ![]() 320 C.E.) gave a much shorter proof of the first conclusion, but it is also conceptually more difficult. Very few of the propositions in the Elements are known by names. Whether this name is due to its difficulty (which it isn’t) or the resemblance of its figure to a bridge is not clear. This proposition has been called the Pons Asinorum, or Asses’ Bridge. It may appear that I.7 only depends on the first conclusion of I.5, but a case of I.7 that Euclid does not discuss relies on the second conclusion of I.5. ![]() Thus, I.13 cannot be used in the proof of I.5. Unfortunately, such an argument would be circular. Proposition I.13 would be enough, since it implies the sum of angles ABC and FBC equals two right angles, and the sum of angles ACB and GCB also equals two right angles, and so the two sums are equal effectively saying all straight angles are equal. But Euclid doesn’t accept straight angles, and even if he did, he hasn’t proved that all straight angles are equal. From the diagram it looks like it would be easy to prove the second conclusion from the first by simply subtracting the equal angles ABC and ACB the straight angles ABF and ACG, respectively. There are two conclusions for this proposition, first that the internal base angles ABC and ACB are equal, second that the external base angles FBC and GCB are equal. ![]()
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