_If two things coincide throughout their whole extent, they are equal._ THEOREMS ILLUSTRATED. [Illustration: Diagram 29.] DEVELOPMENT LESSON. Do the angles Blue, Red, take up all the space on the line _a b_? Do the angles Blue, Yellow, Red, take up all the space on the line? Do the angles Blue, Yellow, Green, Red, take up all the space on the line? Is there room between any two of the angles to put in another angle? Then are not the angles Blue, Yellow, Green, Red, equal to all the space on the line _a b_? NOTE.—The word _space_, as here used, means _angular space_; and it is indispensable that the teacher impress this fact upon the learner. By means of former lessons, the pupil has learned positively, that an angle is the difference between the directions of two lines; and, impliedly, that the included space has nothing to do with the size of the angle. There cannot, therefore, be much danger that the pupil will imbibe any erroneous notion from this style of expression, which is very much more simple than to say that the difference of direction of two given lines is equal to the difference of direction of two other given lines, which style will be used somewhat later in these lessons. [Illustration: Diagram 30.] PROPOSITION I. THEOREM. DEVELOPMENT LESSON. Are the adjacent angles Green, Red, equal to all the angular space on the line _a b_? Place a paper square corner or right angle on the line _a b_ at the _left_ of _c d_ with its vertex at _c_. It will cover all the angle Green and part of the angle Red up to the line _c d_. Now place another square corner on the line _a b_ to the _right_ of the line _c d_, and with its vertex at the point _c_. It will cover the remaining part of the angle Red, and two edges of the square corners will meet along the line _c d_. Are the two right angles equal to all the angular space on the line _a b_? Then if the two adjacent angles Green, Red, are equal to all the angular space on the line _a b_, and the two right angles are also equal to the same space, what do you infer concerning the _adjacent angles_ and the _two right angles_? What axiom do you apply when you say that the _adjacent_ angles are equal to the _two right angles_? To what _same thing_ did you find two things separately equal? What did you first see equal to it? What did you next see equal to it? Then what did you _find_ true? If the angle Red were smaller, and the angle Green larger, would the adjacent angles still be equal to two right angles? Then,— _Any two adjacent angles are equal to two right angles._ If we draw the straight line _c d_ where the edges of the square corners come together, what kind of angles will _a c d_, _d c b_, be?