Proposition I. An equilateral triangle may be described on a given finite straight line. Proposition IX. A given rectilineal angle may be bisected. Definition 24. Of three-sided figures an equilateral triangle is that which has three equal sides. Proposition IV. If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angle contained by those sides equal the third sides are equal. PROPOSITION XI. PROBLEM. To draw a straight line at right angles to a given straight line, from a given point in the same. GIVEN the straight line AB, and C a point in it; IT IS REQUIRED TO DRAW from C a straight line at right angles to AB. From CB or CB produced cut off CE equal to CD. x. and I. 3. therefore the angle DCF is equal to the angle ECF. Because the angle DCF is equal to the adjacent angle ECF; therefore CF is drawn at right angles to AB. Constr. g. Def. 24. I. 8. Proved. Def. 10. Q. E. F. Proposition III. From the greater of two given straight lines a part may be cut off equal to the less. Proposition I. An equilateral triangle may be described on a given finite straight line. Postulate 1. A straight line may be drawn from any one point to any other point. Definition 24. Of three-sided figures an equilateral triangle is that which has three equal sides. Proposition VIII. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal, the angle which is contained by the two sides of the one shall be equal to the angle contained by the two sides equal to them of the other. Definition 10. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of these angles is called a right angle. COROLLARY TO PROPOSITION XI. By the help of this proposition an attempt has been made to show that two straight lines cannot have a common segment. IE If possible, let AD and AC have a common segment AB. From B draw BE at right angles to AB. Because ABC is a straight line; Нур. therefore the angle CBE is equal to the angle ABE. Constr. & Ax. 11. Because ABD is a straight line; Hyp. therefore the angle DBE is equal to the angle ABE. Constr. & Ax. 11. Because the angles DBE and CBE are each equal to the Therefore the angle DBE is equal to the angle CBE. Axiom 1. Because if AD and AC have a common segment, Therefore AD and AC cannot have a common segment. Proved. Axiom 9. G. Q. E. D. Note:-In order to draw BE at right angles to AB we must produce AB, and assume that there is only one way of producing it, that is assume what we have to demonstrate. |