Then if the adjacent angles Yellow, Blue, are equal to two right angles, and the adjacent angles Blue, Red, are also equal to two right angles, what do you think of the two pairs of adjacent angles, Yellow, Blue, and Blue, Red? If, from the adjacent angles Yellow, Blue, we take away the angle Blue, what remains? If, from the adjacent angles Blue, Red, we take away the same angle Blue, what remains? Then, since the same angle Blue has been taken from equal pairs of adjacent angles, what do you think of the two remainders, Yellow, Red? Suppose the lines _a b_ and _m n_ were so drawn that the angles Yellow, Red, were larger or smaller, would they still be equal to each other? Then,— _All vertical angles are equal to each other._ [Illustration: Diagram 34.] DEMONSTRATION. We wish to prove that _All vertical angles are equal to each other._ Let the straight lines _a b_, _m n_, intersect each other at the point _c_, then will any two vertical angles, as Yellow, Red, be equal to each other. For the adjacent angles Yellow, Blue, are equal to two right angles.[3] Footnote 3: When this comparison is made, let the pupil look at the right angles P and S. The adjacent angles Blue, Red, are also equal to two right angles. Therefore the adjacent angles Yellow, Blue, are equal to the adjacent angles Blue, Red. If, from the adjacent angles Yellow, Blue, we take away the angle Blue, we shall have left the angle Yellow. If, from the adjacent angles Blue, Red, we take away the same angle Blue, we shall have left the angle Red. Therefore the vertical angles Yellow, Red, are equal to each other. TEST QUESTIONS. When you say that the adjacent angles Yellow, Blue, are equal to two right angles, do you know it because you _see_ it, or because you have _proved_ it? How do you know that the adjacent angles Blue, Red, are equal to two right angles? When you say the adjacent angles Yellow, Blue, are equal to the adjacent angles Blue, Red, what axiom do you use? What same thing do you take away from equals? From what equals do you take it away? When you take the angle Blue from the adjacent angles Yellow, Blue, what is the remainder? When you take the same angle Blue from the adjacent angles Blue, Red, what is the remainder? What do you find true of the two remainders? What axiom do you use? [Illustration: Diagram 35.] OTHER METHODS OF DEMONSTRATION. The adjacent angles Yellow, Green, are equal to what? The adjacent angles Green, Red, are equal to what? Then what do you know of the two pairs of adjacent angles Yellow, Green, and Green, Red? From the adjacent angles Yellow, Green, take away the angle Green. What remains?