Book 1 - Proposition 7

Given two straight lines constructed from the ends of a straight line and meeting in a point, there cannot be constructed from the ends of the same straight line, and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each equal to that from the same end.

If possible, given two straight lines line AC and line CB constructed on the straight line AB and meeting at the point C, let two other straight lines line AD and line BD be constructed on the same straight line AB, on the same side of it, meeting in another point D and equal to the former two respectively, namely each equal to that from the same end, so that line AC equals line AD which has the same end A, and line BC equals line BD which has the same end point B.

Join line CD.

Since line AC equals line AD, therefore the angle ACD equals the angle ADC. Therefore the angle ADC is greater than the angle DCB. Therefore the angle CDB is much greater than the angle DCB.

Again, since line BC equals line BD, therefore the angle CDB also equals the angle DCB. But it was also proved much greater than it, which is impossible.

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