From the adjacent angles Green, Red, take the same angle Green. What remains? What do you know of the two remainders? Why? What axiom do you use? In the last lesson, when you proved the vertical angles Yellow, Red, equal to each other, you made use of the angle Blue; now prove the same two angles equal by means of the angle Green. The adjacent angles Blue, Red, are equal to what? The adjacent angles Red, Green, are equal to what? Then what do you know of the two pairs of adjacent angles Blue, Red, and Red, Green? From the adjacent angles Blue, Red, take away the angle Red. What remains? From the adjacent angles Red, Green, take away the same angle Red. What remains? Then what do you know of the two remainders, Blue, Green? Now apply the preceding demonstration to the vertical angles Blue, Green. Prove the vertical angles Blue, Green, equal to each other by means of the angle Yellow. [Illustration: Diagram 36.] TEST LESSON. By means of Fig. A,— 1. Prove that the vertical angles Yellow, Red, are equal to each other, using the angle Green. 2. Prove the same thing, using the angle Blue. 3. Prove that the vertical angles Blue, Green, are equal to each other, using the angle Yellow. 4. Prove the same thing, using the angle Red. By means of Fig. B,— 5. Prove the vertical angles Yellow, Red, equal to each other, using the angle Green. 6. Prove the same thing, using the angle Blue. 7. Prove the vertical angles Green, Blue, equal by means of the angle Red. 8. Prove the same thing by means of the angle Yellow. Go through the preceding eight demonstrations again, calling the angles by their letters instead of by their colors. By means of Fig. C, prove that 9. _a c n_ equals _m c b_, by means of _a c m_. 10. _a c n_ equals _m c b_, by means of _b c n_. 11. _a c m_ equals _n c b_, by means of _a c n_. 12. _a c m_ equals _n c b_, by means of _m c b_. By means of Fig. D, prove that 13. _m c a_ equals _b c n_, by means of _a c n_. 14. _m c a_ equals _b c n_, by means of _m c b_. 15. _m c b_ equals _a c n_, by means of _m c a_. 16. _m c b_ equals _a c n_, by means of _b c n_. [Illustration: Diagram 37.] PROPOSITION III. THEOREM. DEVELOPMENT LESSON. In the above diagram, the lines _a b_, _c d_, are parallel, and are intersected by the line _e f_ at the points _m_ and _n_. The angle Red measures the difference of direction between the line _m b_ and what other line? The angle Yellow measures the difference of direction between the line _n d_ and what other line? Then, as the lines _m b_ and _n d_ are parallel, must there not be the same difference of direction between them and the line _e f_? Then can there be any difference between the angles which measure those equal directions?