National Olympiad Final Round

Question

Triangle ABC has AC = BC. The bisector of angle CAB meets side BC at point D. The difference of the sizes of some two internal angles of triangle ABD is 40^. Find all possibilities of what the size of angle ACB can be. FIGURE_0

Accepted Answer

FIGURE_0 Fig. 16 * If - 2 = 40^ then = 80^, whence ACB = 20^. * The case 2 - = 40^ is impossible since it would imply < 0^. * If (180^ - 32) - 2 = 40^ then = 70^, whence ACB = 40^. * If 2 - (180^ - 32) = 40^ then = 110^, but the base angle of an isosceles triangle cannot be obtuse. * If (180^ - 32) - = 40^ then = 56^, whence ACB = 68^. * If - (180^ - 32) = 40^ then = 88^, whence ACB = 4^.