Transverse waves are waves in which the particles of the medium vibrate perpendicular to the direction of the wave's propagation. In most transverse waves, the particles of the medium move back and forth in a direction perpendicular to the direction of the wave's motion, not in a perpendicular direction.
For example, in a water wave, the particles of water vibrate up and down, perpendicular to the direction the wave is moving. In a transverse wave on a string, the particles of the string vibrate back and forth perpendicular to the direction the wave is moving.
There are some exceptions to this general rule. For example, in a surface wave, the particles of the medium move in an elliptical path, with the major axis of the ellipse perpendicular to the direction of the wave's propagation.
However, in most cases, the statement "matter in a medium moves perpendicular to a transverse wave" is not true.
