As the line _d f_ is equal to the line _a c_, upon what point will the point _f_ fall? Then, if the point _e_ falls upon the point _b_, and the point _f_ upon the point _c_, where will the line _e f_ 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. What three parts of the triangle _a b c_ did we suppose to be equal to three corresponding parts of the triangle _d e f_ before we placed one upon the other. What line of the one do we _find_ equal to a line in the other? What two angles of the one do we _find_ equal to two angles in the other? What do you think of the areas of the triangles? [Illustration] DEMONSTRATION. We wish to prove, that, _If two triangles have two sides, and the included angle of the one equal to two sides and the included angle of the other, each to each, the two triangles are equal in all respects._ Let the triangles _a b c_ and _d e f_ have the side _a b_ of the one equal to the side _d e_ of the other; the side _a c_ of the one equal to the side _d f_ of the other; and the included angle _b a c_ of the one equal to the included angle _e d f_ of the other, each to each; then will the two triangles be equal in all their parts. For, place the triangle _d e f_ upon the triangle _a b c_, so that the line _d e_ shall fall upon the line _a b_, with the point _d_ upon the point _a_. Because the line _d e_ is equal to the line _a b_, the point _e_ will fall upon the point _b_. Because the angle _e d f_ is equal to the angle _b a c_, the line _d f_ will fall upon the line _a c_. Because the line _d f_ is equal to the line _a c_, the point _f_ will fall upon the point _c_. Then, because the point _e_ is on the point _b_, and the point _f_ on the point _c_, the line _e f_ will coincide with the line _b c_, and the two triangles will be found equal in all their parts; That is, the angle _e_ is found to be equal to the angle _b_, the angle _f_ to the angle _c_, the line _e f_ to the line _b c_, and the area of the triangle _a b c_ to the area of the triangle _d e f_. [Illustration] PROPOSITION XIV. THEOREM. 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.