Then what do you think of the opposite exterior and interior angles Red, Yellow? DEMONSTRATION. We wish to prove that _Opposite exterior and interior 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 opposite exterior and interior angles, as Red, Yellow, be equal to each other. For the angle Red measures the difference of direction of the lines _m b_ and _e f_. And the angle Yellow measures the difference of direction of the lines _n d_ and _e f_. But because the lines _m b_, _n d_, are parallel, these differences are equal. Therefore the angles which measure them are equal; that is, The opposite exterior and interior angles Red, Yellow, are equal to each other. [Illustration: Diagram 38.] TEST LESSON. By means of Fig. A,— 1. Prove that the opposite exterior and interior angles Green, Blue, are equal to each other. 2. Prove that the opposite exterior and interior angles Red, Yellow, are equal to each other. 3. Prove the opposite exterior and interior angles _c n e_, _a m n_, equal. 4. Prove the opposite exterior and interior angles _e n d_, _n m b_, equal. By means of Fig. B,— 5. Prove the opposite exterior and interior angles _e m a_, _m n d_, equal. 6. Prove the opposite exterior and interior angles _a m n_, _d n f_, equal. 7. Prove the opposite exterior and interior angles _e m b_, _m n c_, equal. 8. Prove the opposite exterior and interior angles _b m n_, _c n f_, equal. [Illustration: Diagram 39.] PROPOSITION IV. THEOREM. DEVELOPMENT LESSON. What do you know of the opposite exterior and interior angles Red, Yellow? What do you know of the vertical angles Red, Green? Then if the interior alternate angles Green, Yellow, are separately equal to the angle Red, what new fact do you know? What axiom do you employ? To what same thing did you find two things equal? What two things did you find equal to it? DEMONSTRATION. We wish to prove that _Any two interior alternate angles are equal to each other._ 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 alternate angles, as Green, Yellow, be equal to each other. For the opposite exterior and interior angles Red, Yellow, are equal. The vertical angles Red, Green, are also equal. Then because the interior alternate angles Green, Yellow, are separately equal to the angle Red, they are equal to each other. [Illustration: Diagram 40.] TEST LESSON. What do you know of the vertical angles Green, Red, in Fig. A?