Mirrors

Abstract

Figuratively, we can slide a ruler along its line and turn the ruler around. This is the idea behind the Ruler-Placement Theorem (Theorem 6.8). Figuratively, we can slide a protractor around a point and turn the protractor over. We don’t have a Protractor-Placement Theorem because this idea is already contained in the Protractor Postulate. (We can start our protractor with any ray \( \overrightarrow {VA} \) and use either halfplane of \( \overleftrightarrow {VA} \).) Our intuition may be galloping ahead of our theory. Nothing has been said about picking up a ruler for line l and putting the ruler down on line m if m ≠ l; nothing has been said about picking up a protractor with initial ray \( \overrightarrow {VA} \) and putting the protractor down on \( \overrightarrow {WB} \) if W ≠ V. Nothing has been said about such things for a very good reason. the Ruler Postulate concerns itself with only one line at a time; the Protractor Postulate concerns itself with only one point at a time.