We are given the following diagram and asked to find the value of x.
From the diagram we can see that ∠ L intercepts arc MP. By the Measure of an Inscribed Angle Theorem, this means the measure of ∠ L is half the measure of MP.
m∠ L=1/2mMPLet's substitute m∠ L with 73^(∘) and find the value of mMP.
Now we can find the measure of the central angle ∠ MOP, which intercepts arc MP.
Let's recall that the measure of a central angle is equal to the measure of its intercepted arc. Therefore, the measure of ∠ MOP is also 146^(∘).
m∠ MOP=146^(∘)
From the diagram, we can also see that the sides of ∠ N are tangent to the circle, so ∠ N is a circumscribed angle. We can use the Circumscribed Angle Theorem, which states the following.
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 ∠ N and the central angle ∠ MOP intercept the same arc MP, so the following is true.
m∠ N=180^(∘)-m∠ MOP
Let's substitute m∠ N with x^(∘) and m∠ MOP with 146^(∘) to find the value of x.
