Proposition 7

The seventh proposition proves that a triangle can be uniquely defined by a line segment, a side of that line segment, and two side lengths. The way Euclid states it is that if two straight lines are constructed off a straight line meeting at a point, there cannot be another pair of straight lines constructed off the same side of that line segment so that the two line segments formed are of equal length to the line segments formed by the original pair of straight lines, but the pair they meet at a different point than the original pair of straight lines. This seems a bit complicated to appear so early in the Elements, but its key to proving SSS congruence in the next proposition. Again, the word "cannot" suggests that this is a proof by contradiction, and that we should assume this "cannot" is a "can". By the assumption, AD and BD are of equal length to AC and BC respectively, and this means that the red angles and yellow angles are congruent. We've already reached a contradiction, because the yellow angle on C is less than the red angle on C, but the yellow angle on D is greater than the red angle on D. The yellow angle can't be both greater than and less than the red angle, so the assumption must have been false.