Commentary on Proposition VI.15¶
Back to Proposition VI.15.
As indicated in the partial converse given in the last note, this proposition is equally true if the angle included by the two sides in one triangle is supplementary, instead of being equal, to the angle included by the two sides in the other.
Let ABC, ADE be two triangles such that the angles BAC, DAE are supplementary, and also
In this case we can place the triangles so that CA is in a straight line with AD, and AB lies along AE (since the angle EAC, being supplementary to the angle EAD, is equal to the angle BAC). If we join BD, the proof given by Euclid applies to this case also.
It is true that VI. 15 can be immediately inferred from VI. 14, since a triangle is half of a parallelogram with the same base and height. But, Euclid’s object being to give the student a grasp of methods rather than results, there seems to be no advantage in deducing one proposition from the other instead of using the same method on each.