 In figure 6.40, angle X is equal to 62 degree, angle XYZ is equal to 54 degree, if YO and ZO are bisectors of angle XYZ and angle XZY respectively of triangle XYZ find angle OZY and angle YOZ. Let us now begin with the solution in triangle XYZ, sum of angles XYZ plus angle YZX plus angle ZXY is equal to 180 degree since sum of three angles of a triangle is equal to 180 degree which is the angle sum property of a triangle. Now angle XYZ is equal to 54 degree plus angle YZX plus angle ZXY which is 62 degree given to us is equal to 180 degree. This implies angle YZX is equal to 180 degree minus 54 degree plus 62 degree which is further equal to 180 degree, sum of 54 and 62 is 116 degree which is further equal to 64 degree thus angle YZX is equal to 64 degree. Let us number this equation as 1. Now since YO is bisector of angle XYZ this implies angle OYZ is equal to half of angle XYZ since a bisector divides the angle into two equal parts therefore angle OYZ will be half of 54 degree which is equal to 27 degree. So angle OYZ is 27 degree and now let us find angle OZY and again since ZO is bisector of angle YZX this implies angle OZY is half of angle YZX which implies it is half of angle 64 degree. This implies that angle OZY is equal to 32 degree. Now we have to find angle YOZ so we will apply the angle some property in this triangle and two of the angles we know so we will find the third angle and now let us consider the small triangle. So in triangle OYZ again the sum of angles OYZ angle YZO plus angle ZOY will be equal to 180 degree since sum of three angles of a triangle is 180 degree which is the angle some property of your book. Now angle OYZ is 27 degree plus angle YZO is 32 degree plus angle ZOY we have to find out so as it is equal to 180 degree or on an angle 27 by 32 we have 59 degree plus angle ZOY is equal to 180 degree and the ZOY is equal to 180 degree minus 59 degree which is further equal to 121 degree. So our answers are angle OZY is equal to 32 degree OY we can also say that angle YOZ is equal to 121 degree so this completes the solution hope you enjoyed it take care and bye for now.