What do you know of the opposite exterior and interior angles Red, Yellow? Then if the interior alternate angles Green, Yellow, are separately equal to the angle Red, what do you infer? By means of Fig. A,— 1. Prove that the interior alternate angles Green, Yellow, are equal, using the angle Red. 2. Prove the same angles equal, using the angle Blue. 3. Go through the same demonstrations again, calling the angles by their letters instead of by their colors. By means of Fig. B,— 4. Prove the interior alternate angles Red, Blue, equal, using the angle Yellow. 5. Prove the same angles equal, using the angle Green. 6. Go through the same two demonstrations again, naming the angles by their letters instead of by their colors. By means of Fig. C,— 7. Prove the interior alternate angles _c n m_, _n m b_, equal, using the angle _f n d_. 8. Prove the same, using the angle _a m e_. 9. Prove the interior alternate angles _a m n_, _m n d_, equal, using the angle _e m b_. 10. Prove the same, using the angle _c n f_. [Illustration: Diagram 41.] PROPOSITION V. THEOREM. DEVELOPMENT LESSON. What do you know of the opposite exterior and interior angles Red, Yellow? What do you know of the vertical angles Yellow, Green? Then if the exterior alternate angles Red, Green, are separately equal to the angle Yellow, what new thing do you know to be true? What axiom do you employ? To what same thing did you know two things to be equal? What two things did you know to be equal to it? Then what new thing did you _find_ to be true? DEMONSTRATION. We wish to prove that _Any two exterior alternate angles are equal to each other._ Let the straight line _e f_ intersect the two parallel straight lines _a b_, _c d_, at the points _m_ and _n_. Then will any two exterior alternate angles, as Red, Green, be equal. For the opposite exterior and interior angles Red, Yellow, are equal to each other. And the vertical angles Yellow, Green, are also equal to each other. Then because the exterior alternate angles Red, Green, are separately equal to the angle Yellow, they are equal to each other. [Illustration: Diagram 42.] TEST LESSON. What do you know of the opposite exterior and interior angles Yellow, Red? What do you know of the vertical angles Red, Blue? Then if the exterior alternate angles Yellow, Blue, are separately equal to the angle Red, what do you know of them? By means of Fig. A,— 1. Prove that the exterior alternate angles Yellow, Blue, are equal, using the angle Red. 2. Prove the same thing, using the angle Green. 3. Go through the same demonstrations, calling the angles by their letters. 4. Prove the exterior alternate angles _e m b_, _c n f_, equal, using the angle _a m n_.