
Here is my explanation for the alternate interior angle.
The lines on opposite each other are parallel to each other. We know that angles on a straight line add up to 180 degrees. If we know that angle DZY is 40 degrees, it means that angle BYZ is 140 degrees. Angle BYZ is on a straght line so angle AYZ should also be 40 degrees since it add up to 180 degrees.
