See now if you can understand the following demonstration:— DEMONSTRATION. We wish to prove that _Any two adjacent angles are equal to two right angles._ Let the two straight lines _a b_, _m n_, intersect each other in the point _c_. (DIAGRAM 30.) Then will any two adjacent angles, as Green, Red, be equal to two right angles? For, from the point _c_, draw the straight line _c d_ so as to make the angles _a c d_, _d c b_, right angles. The adjacent angles Green, Red, are equal to all the angular space on the line _a b_. The right angles _a c d_, _d c b_, are also equal to all the angular space on the line _a b_. Therefore the adjacent angles Green, Red, are equal to two right angles. TEST QUESTIONS. To what same thing did you find two things equal? What did you first see equal to it? What did you next see equal to it? Then what new thing did you find true? What axiom did you make use of? [Illustration: Diagram 31.] TEST LESSON. By means of Fig. A,— 1. Prove that the adjacent angles Green, Red, are equal to two right angles. 2. Prove that the adjacent angles Blue, Yellow, are equal to two right angles. By means of Fig. B,— 3. Prove that the adjacent angles Green, Red, are equal to two right angles. 4. Prove that the adjacent angles Yellow, Blue, are equal to two right angles. By means of Fig. C,— 5. Prove that the adjacent angles Red, Blue, are equal to two right angles. 6. Prove that the adjacent angles Green, Yellow, are equal to two right angles. 7. Give the preceding demonstrations again, but name the angles by their letters instead of by their colors. [Illustration: Diagram 32.] TEST LESSON. By means of Fig. A prove,— 1. That the adjacent angles _a c m_, _m c b_, are equal to two right angles. 2. That the adjacent angles _a c n_, _n c b_, are equal to two right angles. By means of Fig. B prove,— 3. That the adjacent angles _a c n_, _n c b_, are equal to two right angles. 4. That the adjacent angles _a c m_, _m c b_, are equal to two right angles. By means of Fig. C prove,— 5. That the adjacent angles _a c m_, _m c b_, are equal to two right angles. 6. That the adjacent angles _a c n_, _n c b_, are equal to two right angles. By means of Fig. D prove,— 7. That the adjacent angles _a c n_, _n c b_, are equal to two right angles. 8. That the adjacent angles _b c m_, _m c a_, are equal to two right angles. [Illustration: Diagram 33.] PROPOSITION II. THEOREM. DEVELOPMENT LESSON. What kind of angles are P and S? How do the adjacent angles Yellow, Blue, compare with the right angles P, S? How do the adjacent angles Blue, Red, compare with the two right angles?