Given that the measure of angle 𝐵𝐴𝐶 is equal to 𝑥 plus 15 degrees, find 𝑥.
Firstly, let’s look at this angle 𝐵𝐴𝐶 more closely. It’s an angle formed inside a circle with its vertex on the circle itself. The two lines that form the angle — the lines 𝐴𝐵 and 𝐴𝐶 — are both chords of the circle. Therefore, this angle 𝐵𝐴𝐶 is what’s known as an inscribed angle.
Now, we want to find the value of 𝑥. And in order to do so, we need to know the measure of the angle 𝐵𝐴𝐶. Let’s think about how to do this. We’re told the measure of the arc 𝐵𝐶 is 118 degrees. In order to find the measure of the angle 𝐵𝐴𝐶, we need to record a relationship that exists between the measures of inscribed angles and the measures of the arcs that subtend them.
The relationship is this: the measure of an inscribed angle is half the measure of the arc that subtends it. So in our question, this means that the measure of the angle 𝐵𝐴𝐶 is equal to half of the measure of the minor arc 𝐵𝐶.
Now, let’s substitute the expressions or values for these two measures. We have that 𝑥 plus 15 is equal to a half multiplied by 118. This is an equation that we can now solve in order to find the value of 𝑥. Simplifying the right-hand side of the equation gives 𝑥 plus 15 is equal to 59. To solve the equation, we need to subtract 15 from each side. This gives 𝑥 is equal to 44. So 44 is our answer to the problem.
The key fact we used in this question is that the measure of an inscribed angle is half the measure of the arc that subtends it.