PROPOSITION IV. THEOREM. If a straight line meets two other straight lines at a common point, making the sum of the contiguous angles equal to two right angles, the two lines met form one and the same straight line. Let DC meet AC and BC at C, making the sum of the angles DCA and DCB equal to two right angles: then is CB the prolongation of AC. A -B E For, if not, suppose CE to be the prolongation of AC; then is the sum of the angles DCA and DCE equal to two right angles (P. I.): consequently, we have (A. 1), DCA + DCB = DCA + DCE; Taking from both the common angle DCA, there remains DCB = DCE, which is impossible, since a part can not be equal to the whole (A. 8). Hence, CB must be the prolongation of AC; which was to be proved. PROPOSITION V. THEOREM. If two triangles have two sides and the included angle of the one equal to two sides and the included angle of the other, each to each, the triangles are equal in all respects. In the triangles ABC and DEF, let AB be equal to DE, AC to DF, and the angle A to the angle D: then are the triangles equal in all respects. For, let ABC be applied to DEF, in such a manner that the angle A shall coincide with the angle D, the side AB taking the direction DE, and the side AC the AA F direction DF. Then, because AB is equal to DE, the vertex B will coincide with the vertex E; and because AC is equal to DF, the vertex C will coincide with the vertex F; consequently, the side BC will coincide with the side EF (A. 11). The two triangles, therefore, coincide throughout, and are consequently equal in all respects (I., D. 15); which was to be proved. PROPOSITION VI. THEOREM. If two triangles have two angles and the included side of the one equal to two angles and the included side of the other, each to each, the triangles are equal in all respects. In the triangles ABC and DEF, let the angle B be equal to the angle E, the For, let ABC be applied to DEF in such a manner that the angle B shall coincide with the angle E, the side BC taking the direction EF, and the side BA the direc tion ED. Then, because BC is equal to EF, the vertex C will coincide with the vertex F; and because the angle C is equal to the angle F, the side CA will take the direction FD. Now, the vertex A being at the same time on the lines ED and FD, it must be at their intersection D (P. III., C.): hence, the triangles coincide throughout, and are therefore equal in all respects (I., D. 15); which was to be proved. The sum of any two sides of a triangle is greater than the third side. Let ABC be a triangle: then will the sum of any two sides, as AB, BC, be greater than the third side AC. For, the distance from A to C, measured on any broken line AB, BC, is greater than C the distance measured on the straight line AC (A. 12): hence, the sum of AB and BC is greater than AC; which was to be proved. Cor. If from both members of the inequality, ACAB+ BC, we take away either of the sides AB, BC, as BC, for example, there remains (A. 5), that is, the difference between any two sides of a triangle is less than the third side. Scholium. In order that any three given lines may rep resent the sides of a triangle, the sum of any two must be greater than the third, and the difference of any two must be less than the third. PROPOSITION VIII. THEOREM. if from any point within a triangle two straight lines are drawn to the extremities of any side, their sum is less than that of the two remaining sides of the triangle. A Let O be any point within the triangle BAC, and let the lines OB, OC, be drawn to the extremities of any side, as BC: then the sum of BO and OC is less than the sum of the sides BA and AC. B D C Prolong one of the lines, as BO, till it meets the side AC in D; then, from Prop. VII., we have, OC < OD + DC; adding BO to both members of this inequality, recollecting that the sum of BO and OD is equal to BD, we have (A. 4), BO + OC < BD + DC. From the triangle BAD, we have (P. VII.), BDBA + AD; adding DC to both members of this inequality, recollecting that the sum of AD and DC is equal to AC, we have, BD + DC < BA + AC. But it was shown that BO+ OC is less than BD + DC; still more, then, is BO+OC less than BA + AC; which was to be proved. If two triangles have two sides of the one equal to two sides of the other, each to each, and the included angles unequal, the third sides are unequal; and the greater side belongs to the triangle which has the greater included angle. In the triangles BAC and DEF, let AB be equal to DE, AC to DF, and the angle A greater than the angle D: then is BC greater than EF. Let the line AG be drawn, making the angle CAG equal to the angle D (Post. 7); make AG equal to DE, and draw GC. Then the triangles AGC and DEF have two sides and the included angle of the one equal to two sides and the included angle of the other, each to each; consequently, GC is equal to EF (P. V.). Now, the point G may be without the triangle ABC, it may be on the side BC, or it may be within the triangle ABC. Each case will be considered separately. whence, by addition, recollecting that the sum of Bl and IC is equal to BC, and the sum of GI and IA, to GA, we have, AG + BC > AB + GC. |