Exercise 7 Page 565

We are given the following diagram and asked to find the value of $x.$

From the diagram, we can see that $\angle T$ intercepts arc $\overgroup{PQ},$ so by the Measure of an Inscribed Angle Theorem the measure of $\angle T$ is half the measure of $\overgroup{PQ}.$ Let's substitute $m\angle T$ with $x^{\circ }$ and find the value of $m\overgroup{PQ}.$ Now, we can find the measure of the central angle $\angle PRQ,$ which intercepts the arc $\overgroup{PQ}.$ Recall that the measure of a central angle is equal to the measure of its intercepted arc. Therefore, the measure of $\angle PRQ$ is also $2x^{\circ }.$ From the diagram we can also see that the sides of $\angle S$ are tangent to the circle, so $\angle S$ is a circumscribed angle. This means we can use the Circumscribed Angle Theorem. Let's recall what it states!

Circumscribed Angle Theorem

The measure of a circumscribed angle is equal to $180^{\circ }$ minus the measure of the central angle that intercepts the same arc.

The circumscribed angle $\angle S$ and the central angle $\angle PRQ$ intercept the same arc $\overgroup{PQ},$ so the following is true. Let's substitute $m\angle S$ with $50^{\circ }$ and $m\angle PRQ$ with $2x^{\circ }$ and find the value of $x.$ We got that the value of $x$ is $65.$