To solve for the measures of angles $\angle R$, $\angle S$, and $\angle T$ in quadrilateral $QRST$, we need to use the fact that the sum of the interior angles in any quadrilateral is $360^\circ$.

Given:

• $m\angle Q = 68^\circ$
• $m\angle R = (3x + 40)^\circ$
• $m\angle T = (5x - 52)^\circ$

Let $m\angle S = y$.

According to the sum of the interior angles of a quadrilateral: $m\angle Q + m\angle R + m\angle S + m\angle T = 360^\circ$

Substitute the known values: $68 + (3x + 40) + y + (5x - 52) = 360$

Simplify and combine like terms: $(8x + y) + 56 = 360$