Collision in two dimension follows the same momentum conservation principle as motion in one dimension (provided no external force acts on the system and it is isolated). If the collision is elastic in nature then the kinetic energy is also preserved.
If you analyze the motion along X and Y axis, what you will find is that the momentum is conserved along these axis as well. Again this assumes that there is no external force along the dimensions.
Watch this video to develop a better understanding of the topic.
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