Question 377366
Let the point of intersection of XY and ZM be B.  Then from the given, Angle XBZ and Angle YBZ are right angles and supplementary. Hence triangle ZBX and triangle ZBY are right triangles. By the HA Theorem for right triangles, triangle ZBX and triangle ZBY are congruent, and so ZX is congruent to ZY.  Hence triangle YZX is an isosceles triangle.