Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/288

264 never meet EF which lies between them, they cannot meet one another.

I. 32. The corollaries to I. 32 were added by Simson. In the second corollary it ought to be stated what is meant by an exterior angle of a rectilineal figure. At each angular point let one of the sides meeting at that point be produced; then the exterior angle at that point is the angle contained between this produced part and the side which is not produced. Either of the sides may be produced, for the two angles which can thus be obtained are equal, by I. 15.

The rectilineal figures to which Euclid confines himself are those in which the angles all face inwards; we may here however notice another class of figures. In the accompanying diagram the angle AFC faces, outwards, and it is an angle less than two right angles; this angle however is not one of the interior angles of the figure AEDCF. We may consider the corresponding interior angle to be the excess of four right angles above the angle AFC; such an angle, greater than two right angles, is called a re-entrant angle.

The first of the corollaries to I. 32 is true for a figure which has a re-entrant angle or re-entrant angles; but the second is not.

I. 32. If two triangles have two angles of the one equal to two angles of the other each to each they shall also have their third angles equal. This is a very important result, which is often required in the Elements. The student should notice how and 2 one pair of right angles is equal to any other pair of right angles. Then, by I. 32, the three angles of one triangle aro together equal to the three angles of any other triangle. Then, by Axiom 2, the sum of the two angles of one triangle is equal to the sum of the two equal angles of the other; and then, by Axiom 3, the third angles are equal.

After I. 32 we can draw a straight line at right angles to a given straight line from its extremity, without producing the given straight line.

Let AB be the given straight line. It is required to draw from A a straight line at right angles to AB.