If the two objects are brought into thermal contact, object B gives up more heat and changes its temperature more than object A. The heat released by one object is absorbed by the other, and the temperature change of each is the same because the specific heats are the same.
When two objects with different temperatures come into contact with one another, energy moves from the hotter (higher temperature) object to the cooler (lower temperature) object until both objects reach the same temperature.
The thermal energy of the object increases as the average kinetic energy of its constituent particles rises. Therefore, as an object's temperature rises, so does its thermal energy.
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