We are given the following diagram and asked to find the value of x.
From the diagram, we can see that ∠ T intercepts arc PQ, so by the Measure of an Inscribed Angle Theorem the measure of ∠ T is half the measure of PQ.
m∠ T=1/2mPQLet's substitute m∠ T with x^(∘) and find the value of mPQ.
Now, we can find the measure of the central angle ∠ PRQ, which intercepts the arc PQ.
Recall that the measure of a central angle is equal to the measure of its intercepted arc. Therefore, the measure of ∠ PRQ is also 2x^(∘).
m∠ PRQ=2x^(∘)
From the diagram we can also see that the sides of ∠ S are tangent to the circle, so ∠ 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^(∘) minus the measure of the central angle that intercepts the same arc.
The circumscribed angle ∠ S and the central angle ∠ PRQ intercept the same arc PQ, so the following is true.
m∠ S=180^(∘)-m∠ PRQ
Let's substitute m∠ S with 50^(∘) and m∠ PRQ with 2x^(∘) and find the value of x.
