Therefore there is a quadrilateral Figure &c. Q.E.D. Definition. [Pg 23] A rectangular quadrilateral Figure may be called a 'Rectangle.' PROP. VI. Theorem. [Pg 24] The opposite sides of a Rectangle are equal. Let be a Rectangle; and let it be reversed so that , , may change places. Then will lie along , and along . Now, if were not equal to , , , would not change places, but would take new positions, as ; hence exterior angle would be greater than interior opposite angle ; [Euc. I. 18. but they are also equal, being right angles; which is absurd; . Similarly it may be proved that . Therefore the opposite sides &c. Q.E.D. PROP. VII. Theorem. There is a Pair of Lines, each of which is 'equidistant' from the other, that is, is such that all Points on it are equally distant from the other Line. Let be a Rectangle; and let a vertical Line be supposed, first to coincide with , and then to move along , continuing always at right angles to it, till it reaches some intermediate position . Now, if its top be not now on , it must have either dropped below it or risen above it. First, let it be supposed to have dropped below it: and join , . Then, if the Figure be reversed, and applied to the same base, it is evident that and will exchange places; angle , i.e. it is less than ; Similarly angle is less than ; but angle is less than . angles at are together less than ; which is absurd; [Euc. I. 13. Cor. top of vertical Line has not dropped below . Similarly it may be proved that it has not risen above . Hence it moves along ; i. e. it describes a straight Line, and will evidently continue to do so, however far the vertical Line move, either way, along . Therefore there is a Pair of Lines &c. Q.E.D. Corollaries. [Pg 25] 1. If a Pair of Lines have a common perpendicular: each of them is equidistant from the other. Corollary 2. [Pg 26] It is possible to form a Rectangle of any given width and height. For, in the above Pair of horizontal equidistant Lines, 2 common perpendiculars may be drawn, at a given width apart; and the Figure, so formed, will be a Rectangle; and its sides will be a Pair of vertical equidistant Lines, which may be treated in the same way. PROP. VIII. Theorem. The angles of any Triangle are together equal to two right angles. Let be a Triangle, so placed that each base-angle is acute; from draw perpendicular to ; and make Rectangles , . [II. Prop. 7. Cor. 2. has its opposite sides equal, [II. Prop. 6. , in Triangles , , the sides of the one are respectively equal to the sides of the other; ; [Euc. I. 8. Similarly angle ; the angles of the Triangle together = the angles , , ; i. e. together = . Therefore the angles of any Triangle &c. Q.E.D. PROP. IX. Theorem. [Pg 27] A Pair of Lines, which are equally inclined to a certain transversal, are so to any transversal. Let , , be equally inclined to transversal ; and let be any other transversal. Join .