1. **State the problem:** We need to find the size of angle $t$ in a quadrilateral where two angles are given as $144^\circ$ and $164^\circ$, and the figure has two parallel sides. 2. **Recall the rule:** In a quadrilateral with two parallel sides, the consecutive interior angles between the parallel lines are supplementary, meaning they add up to $180^\circ$. 3. **Identify the pairs:** The angle $144^\circ$ and the angle adjacent to $t$ on the same parallel line add up to $180^\circ$. 4. **Calculate the adjacent angle to $t$:** $$180^\circ - 144^\circ = 36^\circ$$ 5. **Use the fact that the sum of angles around a point is $360^\circ$:** The angles around the point where $t$ is located are $t$, $36^\circ$, and $164^\circ$. 6. **Set up the equation:** $$t + 36^\circ + 164^\circ = 360^\circ$$ 7. **Simplify:** $$t + 200^\circ = 360^\circ$$ 8. **Solve for $t$:** $$t = 360^\circ - 200^\circ = 160^\circ$$ **Final answer:** $$\boxed{160^\circ}$$