a In order to find the measure of BC let's use the Inscribed Angle Theorem, which states the following.
Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of its intercepted arc.
Analyzing the diagram, we can see that arc BC is intercepted by the inscribed angle A, which measures 48^(∘).
From the theorem, we know the following.
m∠ A=1/2m BC
Multiplying both sides of the equation by 2, we get that the measure of the arc is twice greater that the measure of ∠ A.
2m∠ A=m BC
Let's substitute m∠ A with 48^(∘) and calculate m BC.
m BC=2m∠ A=2( 48^(∘))=96^(∘)
b We are asked to find the measure of angle ∠ B. Let's examine the given diagram.
As we can see, angle ∠ B intercepts the arc BC, which measures 100^(∘). Now, we can use the Inscribed Angle Theorem again.
Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of its intercepted arc.
According to the theorem, the measure of ∠ B is half the measure of the intercepted arc AC.
m∠ B=1/2mAC
Let's substitute mAC with 110^(∘) and calculate m∠ B.
m∠ B=1/2( 110^(∘))=55^(∘)
c To find the measure of ∠ C, let's analyze the triangle △ ABC.
We are given that m∠ A=48^(∘) and from Part B we know that m∠ B=55^(∘). By the Triangle Angle Sum Theorem the sum of the angles' measures in any triangle is 180^(∘). Applying it to △ ABC, we can write the following equation.
m∠ A+m∠ B+m∠ C=180^(∘)
Let's substitute the known measures and find m∠ C.
d Here, we are asked to find the measure of the arc AB. Once again, let's consider the given diagram.
From Part C, we know that angle ∠ C measures 77^(∘) and that is the angle that intercepts arc AB. By the Inscribed Angle Theorem we can conclude the following.
m∠ C=1/2m AB
Let's substitute m∠ C with 77^(∘) and solve the equation for m AB.
