THEOR. 6. If two triangles have two angles of the one equal to two angles of the other, each to each, and have likewise the sides between the vertices of these angles equal, then the triangles are identically equal, and of the sides those are equal which are opposite to the equal angles. Let ABC, DEF be two triangles having the angle ABC equal to the angle DEF, the angle ACB to the angle DFE, and the side BC to the side EF: then shall the triangles be identically equal, having the angle BAC equal to the angle EDF, the side AC to the side DF, and the side AB to the side DE. Let the triangle ABC be applied to the triangle DEF, so that the point B may fall on the point E, the side BC along the side EF, and the point A on the same side of EF as the point Ꭰ ; then C will fall on F, since BC is equal to EF, Нур. BA will fall along ED, since the angle CBA is equal to the angle FED Нур. and CA will fall along FD, since the angle BCA is equal to the angle EFD; Нур. hence A, which is the point of intersection of BA and CA, will fall on D, which is the point of intersection of ED and FD, Ax. I. and the triangle ABC will coincide with the triangle DEF, and is therefore identically equal to it, the angle BAC equal to the angle EDF, the side AC to the side DF, and the side AB to the side DE. Q.E.D. Ex. 10. If the bisector of an angle of a triangle is also perpendi cular to the opposite side, the triangle is isosceles. THEOR. 7. If two sides of a triangle are equal, the angles opposite to those sides are equal. Let ABC be a triangle having the side AB equal to the side AC: then shall the angle ACB be equal to the angle ABC. Let A'B'C' be a triangle identically equal to the triangle ABC, the points A',B',C' corresponding respectively to the points A,B,C. Then in the triangles ABC, A'C'B', AB is equal to A'C', since it is equal to AC, and AC is equal to A'B', since it is equal to AB, and the angle BAC is equal to the angle C'A'B', Hyp. Нур. . therefore the angle ACB, which is opposite to the side AB, is equal to the angle A'B'C', which is opposite to the side A'C', 1. 5. that is, the angle ACB is equal to the angle ABC. Q.E.D. COR. If a triangle is equilateral, it is also equiangular. Ex. 11. Prove Theor. 7 by comparing the triangles into which the bisector of the vertical angle divides the isosceles triangle. THEOR. 8. If two angles of a triangle are equal, the sides opposite to those angles are equal. Let ABC be a triangle having the angle ABC equal to the angle ACB : then shall the side AC be equal to the side AB. Let A'B'C' be a triangle identically equal to the triangle ABC, the points A'B',C' corresponding respectively to the points A,B,C. Then in the triangles ABC, A'C'B', the angle ABC is equal to the angle A'C'B', since it is equal to the angle ACB, Нур. and the angle ACB is equal to the angle A'B'C', since it is equal to the angle ABC, Нур. and BC is equal to C'B', therefore the side AC, which is opposite to the angle ABC, is equal to the side A'B', which is opposite to the angle A'C'B', 1. 6. that is, the side AC is equal to the side AB. Q.E.D. COR. If a triangle is equiangular, it is also equilateral. Ex. 12. If the angles at the base of an isosceles triangle are bisected, the bisectors and the base form an isosceles triangle. THEOR. 9. If any side of a triangle is produced, the exterior angle is greater than either of the interior opposite angles. Let ABC be a triangle having the side BC produced to D: then shall the exterior angle ACD be greater than either of the interior opposite angles BAC, ABC. Let E be the middle point of AC; join BE, and produce BE to F, making EF equal to BE, join FC. 1. 4. I. 5. Ax. a. Then in the triangles ECF, EAB, the side EC is equal to the side EA, the side EF is equal to the side EB, and the angle CEF is equal to the angle AEB, since they are vertically opposite, therefore the angle ECF is equal to the angle EAB, but the angle ACD is greater than the angle ECF, therefore the angle ACD is also greater than the angle BAC; which is the angle opposite to the side BC which is produced. Again, produce AC to G, then, in like manner, the angle BCG is greater than the angle ABC which is opposite to AC, but the angle BCG is equal to the angle ACD, because they are vertically opposite; therefore the angle ACD is also greater than the angle ABC. Q.E.D. I. 4. Ex. 13. Shew that the sum of the angles of the triangle FBC is equal to the sum of the angles of the triangle ABC. THEOR. 10. Any two angles of a triangle are together less than two right angles. |