Given RST = x − 8^, the measure of arc RT is:

The angle $\angle RST$ is a central angle that intercepts arc RT. Therefore, the measure of arc RT is equal to the measure of $\angle RST$.
The angle $\angle PQR$ is an inscribed angle that intercepts arc PR. Therefore, the measure of arc PR is twice the measure of $\angle PQR$.

Given $\angle RST = x - 8^\circ$, the measure of arc RT is:

Given $\angle PQR = 2x - 40^\circ$, the measure of arc PR is:

Since PT is a diameter, it divides the circle into two semicircles. The arc PRT is a semicircle, which means its total measure is $180^\circ$. Arc PRT is the sum of arc PR and arc RT.

Substitute the expressions for arc PR and arc RT into the equation from Step 3:

Combine like terms:

Add 88 to both sides of the equation:

Divide by 5 to find the value of $x$:

The value of $x$ is $\boxed{\text{53.6}}$.