Let's consider variables a, b, and c one at a time.
Variable a
We will start with a. First, let's analyze the given diagram. We will also name some of the points for future reference.
Let's recall that the measure of an arc is equal to the measure of the central angle that creates it. From the diagram we can see that angle ∠ NOM, which measures 44^(∘), is a central angle of the arc NM. This allows us to conclude the following.
mNM=44^(∘)
Now, let's use the Inscribed Angle Theorem.
Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of its intercepted arc.
According to this theorem, the measure of the angle ∠ a is half the measure of its intercepted arc, NM.
m∠ a=1/2mNM
Let's substitute mNM with 44^(∘) and calculate m∠ a.
m∠ a=1/2( 44^(∘))=22^(∘)
Therefore, the value of a is 22.
Variable c
Since to find the value of b we need to know c, let's calculate it first. Consider the diagram below.
We can see that value of c is the measure of the arc KN. The sum of the arcs around a circle is 360^(∘). By the Arc Addition Postulate, if we subtract from 360^(∘) the measures of KM and NM we can calculate mKN.
mKN=360^(∘)-mKM-mNM
It is given that mKM= 160^(∘), and earlier we have found that mNM= 44^(∘). Let's substitute these values into the equation and solve it for mKN.
mKN=360^(∘)- 160^(∘)- 44^(∘)=156^(∘)
We conclude that the value of c is 156.
Variable b
Finally, let's calculate b. Again, we will start with examining the diagram.
As we can see, ∠ b formed by the tangent to K and chord KN. We can apply Theorem 12-12, which states the following.
Theorem 12-12
The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.
By the theorem, m∠ b is half the measure of its intercepted arc KN.
m∠ b=1/2mKN
We have found that mKN is 156^(∘), so we have enough information to find the measure of ∠ b.
m∠ b=1/2( 156^(∘))=78^(∘)
