Step 1: Don't panic
This is a pretty regular congruent triangles proof. We're given two pairs of angles and a pair of segments are congruent; that's almost enough for Angle Side Angle. What we need to show is the two sides included between two pairs of congruent angles are congruent. That's pretty straightforward.
I'm not going to typeset this; read AB as segment AB, = as congruent and the angles as angle N or angle MON.
YO = NZ Given
YO + OZ = NZ + OZ Equals added to equals are equal
YZ = ON Length between collinear points theorem
angles M=X and N=Y Given
angle MON= YZX Triangles angles add to 180 so if two pairs are congruent all three are
triangle MNO = XYZ Angles Side Angle