The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.
x=75
y=210
w=105
Practice makes perfect
Consider the given diagram.
We are going to find the values of x, y, and w one at a time.
Finding x
Recall that the measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.
In our diagram, the angle whose measure is x^(∘) is formed by a tangent and a chord. Its intercepted arc has a measure of 150^(∘). Therefore, we know that x is half of 150.
x=1/2(150) ⇔ x=75
We have found that x^(∘)=75^(∘).
Finding w
Let's now focus on finding the value of w. The tangent line to the given circle is a straight line.
From this, we know that the angles measuring w^(∘) and x^(∘) are supplementary and therefore add to 180^(∘). We also already found that x^(∘)=75^(∘).
75+w=180 ⇔ w=105
We have found that w^(∘)=105^(∘).
Finding y
Look at the angle of measure w=105, formed by the tangent and the chord. We want to find the measure of the related intercepted arc, y^(∘).
Using the same fact as before, we know that 105^(∘) is half of y^(∘).
105=1/2y ⇔ y=210
We have found that y^(∘)=210^(∘). We can confirm this by remembering that the full turn of a circle equals 360^(∘). We know from the diagram that one arc measures 150^(∘). We can find the measure of the larger arc by subtracting 105^(∘) from 360^(∘).
150+y=360 ⇔ y=210
We found again that y^(∘)=210^(∘).
