Marks' first lessons in geometry / In two parts. Objectively presented, and designed for the use of primary classes in grammar schools, academies, etc.

by Bernhard Marks

DEVELOPMENT LESSON.

In these two triangles we have tried to make the angle b of the one equal to the angle e of the other; the angle c of the one equal to the angle f of the other; and the included side b c of the one equal to the included side e f of the other.

We now wish to find out if the remaining angle a of the one is equal to the remaining angle d of the other, and if the two remaining sides a b and a c of the one are equal to the two remaining sides d e and d f of the other.

Suppose we were to cut the triangle d e f out of the page and place it upon the triangle a b c, so that the line e f shall fall upon the line b c, with the point e upon the point b.

Because the line e f is equal to the line b c, upon what point will the point f fall?

Because the angle e is equal to the angle b, where will the line e d fall?

Because the angle f is equal to the angle c, where will the line d f fall?

Then, if the line d e falls upon the line a b and the line d f upon the line a c, where will the point d fall?

Now because the three sides of the triangle d e f exactly fall upon the three sides of the triangle a b c, we say the two magnitudes coincide throughout their whole extent, and are therefore equal.

Suppose the angle e were greater than the angle b, would the line e d fall within or without the triangle?