Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/45

 and the angle AEB is equal to the angle CEF, because they are opposite vertical angles; therefore the triangle AEB is equal to the triangle CEF, and the remaining angles to the remaining angles, each to each, to which the equal sides are opposite; therefore the angle BAE is equal to the angle ECF. But the angle ECD is greater than the angle ECF. Therefore the angle ACD is greater than the angle BAE. In the same manner if BC be bisected, and the side AC be produced to G, it may be shewn that the angle BCG, that is the angle ACD, is greater than the angle ABC. Wherefore, if one side &c. PROPOSITION 17. THEOREM. Any two angles of a triangle are together less than two right angles. Let ABC be a triangle: any two of its angles are together less than two right angles. Produce BC to D. Then because ACD is the exterior angle of the triangle ABC, it is greater than the interior opposite angle ABC. To each of these add the angle ACB Therefore the angles ACD, ACB are greater than the angles ABC ACB. But the angles ACD, ACB are together equal to two right angles. Therefore the angles ABC, ACB are together less than two right angles. In the same manner it may be shewn that the angles BAC, ACB, as also the angles CAB, ABC, are together less than two right angles. Wherefore, any two angles &c.