Euclid’s proof is somewhat complicated, and a stumbling-block to many schoolboys. The proof becomes much simpler if we consider the isosceles triangle ABC (AB = AC) twice over, once as a triangle BAC, and once as a triangle CAB; and now remember that AB, AC in the first are equal respectively to AC, AB in the second, and the angles included by these sides are equal. Hence the triangles are equal, and the angles in the one are equal to those in the other, viz. those which are opposite equal sides, i.e. angle ABC in the first equals angle ACB in the second, as they are opposite the equal sides AC and AB in the two triangles.
There follows the converse theorem (Prop. 6). If two angles in a triangle are equal, then the sides opposite them are equal,—i.e. the triangle is isosceles. The proof given consists in what is called a reductio ad absurdum, a kind of proof often used by Euclid, and principally in proving the converse of a previous theorem. It assumes that the theorem to be proved is wrong, and then shows that this assumption leads to an absurdity, i.e. to a conclusion which is in contradiction to a proposition proved before—that therefore the assumption made cannot be true, and hence that the theorem is true. It is often stated that Euclid invented this kind of proof, but the method is most likely much older.
§ 8. It is next proved that two triangles which have the three sides of the one equal respectively to those of the other are identically equal, hence that the angles of the one are equal respectively to those of the other, those being equal which are opposite equal sides. This is Prop. 8, Prop. 7 containing only a first step towards its proof.
These theorems allow now of the solution of a number of problems, viz.:—
To bisect a given angle (Prop. 9).
To bisect a given finite straight line (Prop. 10).
To draw a straight line perpendicularly to a given straight line through a given point in it (Prop. 11), and also through a given point not in it (Prop. 12).
The solutions all depend upon properties of isosceles triangles.
§ 9. The next three theorems relate to angles only, and might have been proved before Prop. 4, or even at the very beginning. The first (Prop. 13) says, The angles which one straight line makes with another straight line on one side of it either are two right angles or are together equal to two right angles. This theorem would have been unnecessary if Euclid had admitted the notion of an angle such that its two limits are in the same straight line, and had besides defined the sum of two angles.
Its converse (Prop. 14) is of great use, inasmuch as it enables us in many cases to prove that two straight lines drawn from the same point are one the continuation of the other. So also is