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 *AC* and *CB* constructed on the straight line *AB* and meeting at the point *C,* let two other straight lines *AD* and *DB* 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 *AC* equals *AD* which has the same end *A,* and *CB* equals *DB* which has the same end *B.*

Join *CD.*

Since *AC* equals *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 *CB* equals *DB,* therefore the angle *CDB* also equals the angle *DCB.* But it was also proved much greater than it, which is impossible.

Therefore *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.*

Q.E.D.

- ... the angle

This property is not among the listed Common Notions.

Next, transitivity of “less than”

justifies the last statement “*CDB* is much greater than the angle *DCB.*” Transitivity is another property not listed as a Common Notion.

As in the proof of the last proposition and many to come, the law of trichotomy is also used. Here it’s used to reach the final contradiction.