We are asked to find the values of the variables on the diagram. Let's calculate them one at a time, starting with b.
Variable b
Consider the following diagram.
As we can see, angle ∠ b intercepts the arc GF, which measures 120^(∘). Let's recall what the Inscribed Angle Theorem states.
Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of its intercepted arc.
By the theorem, the measure of ∠ b is half the measure of GF. Hence, dividing 120 by 2, we can calculate m∠ b.
m∠ b=120/2=60^(∘)
This way we got that the value of b is 60.
Variable a
Now, we can calculate the value of a. Note that an inscribed angle (a+b)^(∘), which we can call ∠ FHE, is inscribed in a semicircle.
We can apply Corollary 2, which states the following.
Corollary 2
An angle inscribed in a semicircle is a right angle.
It allows us to conclude that ∠ FHE is a right angle. Thus, the sum of a and b is 90^(∘).
a+b=90
We already know that b= 60. Let's substitute this value into the equation and calculate a.
a+ 60=90 ⇒ a=30
Variable e
Now that we know the value of a, we can find e. From the diagram, we can see that e^(∘) is the measure of an arc intercepted by angle of the measure a^(∘).
By the Inscribed Angle Theorem, the measure of ∠ a is half the measure of GE, which means that arc GE measures twice as much as ∠ a.
m∠ a=1/2mGE ⇒ mGE=2m∠ a
Let's substitute a with 30 and calculate mGE.
mGE=2(30)=60^(∘)
We conclude that the value of e is 60.
Variable d
In order to find the value of d, let's use the fact that the measures of arcs around a circle is 360^(∘). This allows us to write the following equation.
56^(∘)+e^(∘)+120^(∘)+d^(∘)=360^(∘)
Earlier we have found that e equals 60. Let's substitute this value and solve the equation for d.
Finally, we can calculate the value of c. We can start with analyzing the diagram.
Angle ∠ c is formed by the tangent and chord FH. Let's use 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.
From the diagram we can see that angle ∠ c intercepts arc FG. Therefore by the theorem its measure is half the measure of FG.
m∠ c=1/2mFG
The measure of arc FG is d^(∘), which we have found to be 124^(∘). Let's substitute this value into the equation and calculate m∠ c.
m∠ c=1/2( 124^(∘))=62^(∘)
Therefore, the value of c is 62.
