1. **Problem statement:** Given kite RSTU with angles labeled as $m\angle R = x + 30^\circ$, $m\angle T = x^\circ$, and $m\angle U = 125^\circ$, and sides RS = RU and ST = UT, find $m\angle R$, $m\angle S$, and $m\angle T$. 2. **Properties of a kite:** In a kite, two pairs of adjacent sides are equal. The angles between unequal sides are equal. Here, RS = RU and ST = UT, so $m\angle S = m\angle T$. 3. **Sum of interior angles:** The sum of angles in any quadrilateral is $360^\circ$. So, $$m\angle R + m\angle S + m\angle T + m\angle U = 360^\circ$$ 4. **Express known angles:** Substitute known values and relations: $$ (x + 30) + m\angle S + x + 125 = 360 $$ Since $m\angle S = m\angle T = x$, rewrite: $$ (x + 30) + x + x + 125 = 360 $$ 5. **Simplify and solve for $x$:** $$ 3x + 155 = 360 $$ $$ 3x = 360 - 155 $$ $$ 3x = 205 $$ $$ x = \frac{205}{3} = 68\frac{1}{3}^\circ $$ 6. **Find each angle:** - $m\angle R = x + 30 = 68\frac{1}{3} + 30 = 98\frac{1}{3}^\circ$ - $m\angle T = x = 68\frac{1}{3}^\circ$ - $m\angle S = m\angle T = 68\frac{1}{3}^\circ$ **Final answers:** $$m\angle R = 98\frac{1}{3}^\circ, \quad m\angle S = 68\frac{1}{3}^\circ, \quad m\angle T = 68\frac{1}{3}^\circ$$