Your kite definition probably states that a kite has two pairs of adjacent congruent sides. In this case SN = NO and OW = WS. Use the diagram to justify, since it is clarifying which sides are congruent. (This really should have been stated among the givens.)

You also can say NW = NW.

Use that to prove two triangles are congruent using SSS congruence.

Then certain corresponding angles must be congruent because of the triangles being congruent. Reason might be "Corresponding parts of congruent triangles are congruent."

Then you need a little more reasoning related to two adjacent angles being equal in measure implying that the ray NW they have in common bisects the angle SNO.

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Minor detail: If your instructor is a stickler for detail, your proof might need to say that two angles being congruent implies their measures are equal (def. of congruent angles), if your definition of angle bisector refers to creating angles with equal measure rather than creating congruent angles.