1. **State the problem:** We need to find the measures of all angles in an isosceles trapezoid where one angle is given as $82^\circ$. 2. **Recall properties of isosceles trapezoids:** In an isosceles trapezoid, the non-parallel sides are equal in length, and the base angles are congruent. Also, consecutive angles between the parallel sides are supplementary (sum to $180^\circ$). 3. **Identify given angle:** Suppose $m\angle S = 82^\circ$ (the given angle on the right side). 4. **Find adjacent angle:** Since $\angle S$ and $\angle R$ are consecutive angles between parallel sides, they are supplementary: $$m\angle S + m\angle R = 180^\circ$$ $$82^\circ + m\angle R = 180^\circ$$ $$m\angle R = 180^\circ - 82^\circ = 98^\circ$$ 5. **Use isosceles property:** The base angles on the other side are congruent to these angles: $$m\angle Q = m\angle R = 98^\circ$$ $$m\angle T = m\angle S = 82^\circ$$ 6. **Final answer:** $$m\angle Q = 98^\circ$$ $$m\angle T = 82^\circ$$ $$m\angle R = 98^\circ$$