Question

triangle XYZ is an isosceles right triangle, right angled at Y.M and N are mid points of XY and XZ respectively. find the measure of angle XNM.

Answer 1

triangle XYZ is an isosceles right triangle, right at Y. M and N are mid points of XY and XZ. so angle ZYX is of 90 degree
so, angle yzx is of 45 degree
and angle yxz is also of 45 degree
so line segment MN will be perpendicular to segment YZ.
thus angle XNM will be of 45 degree

Answer 2

it is 45 deg. = π/4.

Since, XYZ is a right angle triangle  at Y,  angle Y = π/2.
Since it is an isosceles triangle:  angle X = angle Z
 sum of angles X + angle Z = π - angle Y = π - π/2 = π/2 
     hence angle X = angle Z = π/4

Now, the triangle XNM  is also an isosceles right angle triangle.  The reasons:
   1. NM is a line joining the mid points of sides XY and XZ respectively.  Hence NM is parallel to the third side YZ.  also,  the length of NM is 1/2 * YZ

   2.  As  NM  ||  YZ,  the transversal XZ cuts the two parallel lines at Z and N.  The angles XZY and XNM are  equal.  They are included angles on the same side of transversal.    Hence,  angle XNM = angle XZY = π/4.

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further more:
   3. As  NM  || YZ  and  YZ  is perpendicular to XY,    NM is perpendicular to  XY.  Hence  angle at N in triangle XNM is also a right angle.