PROPOSITION XII. THEOREM. DEVELOPMENT LESSON. Let the Fig. A B C D be a parallelogram. Produce the side C D to F. Because the straight line B D intersects the parallels A B and C F, the angle B is equal to what other angle? Because the straight line C F intersects the parallels A C and B D, the angle C is equal to what other angle? Then what follows from this? To what angle did you find two others equal? What two angles did you find equal to it? What axiom do you think of? See if you can go through the demonstration without reading it even once. DEMONSTRATION. We wish to prove that _The opposite angles of a parallelogram are equal to each other._ Let the Fig. A B C D be a parallelogram. Then will any two opposite angles, as B and C, be equal to each other. For produce the line C D to F. Because the straight line B D meets the two parallels A B and C F, The interior alternate angles B and E are equal to each other. Because the straight line C F meets the two parallels B D and A C, The opposite exterior and interior angles C and E are equal to each other. Then, because the angles B and C are separately equal to the angle E, they are equal to each other. * * * * * 1. Prove the same by producing the line A B towards the left. 2. Prove the same by producing the line B D downwards. 3. Prove the angles A and D equal to each other by producing the line C D towards the left. 4. Prove the same by producing the line D B upwards. 5. See if you can prove the same by drawing a diagonal through the points A and D. [Illustration] PROPOSITION XIII. THEOREM. DEVELOPMENT LESSON. In these two triangles we have tried to make 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. We now wish to find out if the third side _b c_ of the one is equal to the third side _e f_ of the other, and if the two remaining angles _b_ and _c_ of the one are equal to the two remaining angles _e_ and _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 _d e_ should fall upon the line _a b_, and the point _d_ upon the point _a_. As the line _d e_ is equal to the line _a b_, upon what point will the point _e_ fall? If the angle _e d f_ were less than the angle _b a c_, would the line _d f_ fall within or without the triangle? If the angle _e d f_ were greater than the angle _b a c_, where would the line _d f_ fall? Since the angle _a_ is equal to _d_, where, then, must the line _d f_ fall?