5. Prove the same, using the angle _m n d_. By means of Fig. B,— 6. Prove that the exterior alternate angles _c m e_, _f n b_, are equal, using the angle _n m d_. 7. Prove the same, using the angle _a n m_. 8. Prove the exterior alternate angles _e m d_, _a n f_, equal, using the angle _c m n_. 9. Prove the same, using the angle _m n b_. [Illustration: Diagram 43.] PROPOSITION VI. THEOREM. DEVELOPMENT LESSON. What do you know of the interior alternate angles Yellow, Red? If to the angle Green you add the angle Yellow, what is the sum? If to the same angle Green you add the equal angle Red, what is the sum? Then, having added equals to the same thing, what do you think of the two sums,—the adjacent angles Green, Yellow, and the interior opposite angles Green, Red? What do you know of the adjacent angles Green, Yellow, and the right angles P, S? Then if the interior opposite angles Green, Red, and the two right angles P, S, are separately equal to the adjacent angles Green, Yellow, what new thing do you know? DEMONSTRATION. We wish to prove that _Any two interior opposite angles are equal to two right angles._ Let the straight line _e f_ intersect the two parallel straight lines _a b_, _c d_, in the points _m_ and _n_. Then will any two interior opposite angles be equal to two right angles. For the interior alternate angles Yellow, Red, are equal. If to the angle Green we add the angle Yellow, we shall have the adjacent angles Green, Yellow. If to the same angle Green we add the equal angle Red, we shall have the interior opposite angles Green, Red. Then the adjacent angles Green, Yellow, are equal to the interior opposite angles Green, Red. But the adjacent angles Green, Yellow, are equal to two right angles. Then because the interior opposite angles Green, Red, and two right angles, are separately equal to the two adjacent angles Green, Yellow, they are equal to each other. [Illustration: Diagram 44.] TEST LESSON. By means of Fig. A,— 1. Prove the interior opposite angles Green, Yellow, equal to two right angles, using the angle Red. 2. Prove the same, using the angle Blue. 3. Prove the same, using the angle _e g b_. 4. Prove the same, using the angle _f h d_. 5. Go through the same demonstrations again, naming the angles by their letters instead of by their colors. 6. Prove the interior opposite angles Red, Blue, equal to two right angles, using the angle Yellow. 7. Prove the same, using the angle Green. 8. Prove the same, using the angle _e g a_. 9. Prove the same, using the angle _c h f_. 10. Go through the same demonstrations again, calling the angles by their letters instead of by their colors. By means of Fig. B,—