Question Video: Finding the Measure of an Angle Given Its Arc’s Measure Using an Inscribed Angle Mathematics

Video Transcript

Suppose that the line 𝑋𝑌 is a tangent to the circle with center 𝑀 at 𝐵, line segment 𝐵𝐴 is parallel to line segment 𝑀𝐶, and the measure of angle 𝐴𝐵𝑌 is 51.8 degrees. Find the measure of angle 𝐶𝐵𝑋.

Let’s begin by marking the angle we’ve been given and the angle we wish to calculate on the diagram. Angle 𝐴𝐵𝑌 is 51.8 degrees, and we’re looking for the measure of angle 𝐶𝐵𝑋. We can now see that the angle whose measure we’ve been asked to calculate is an angle of tangency, as it is the angle between the tangent 𝑋𝑌 and the chord 𝐵𝐶. We can therefore recall a key theorem, which states that the measure of an angle of tangency is equal to half the measure of the central angle subtended by the same arc. The arc connecting the endpoints of the chord 𝐵𝐶 is the minor arc 𝐵𝐶, and the central angle subtended by this arc is angle 𝐵𝑀𝐶. So we have that the measure of angle 𝐶𝐵𝑋 is half the measure of angle 𝐵𝑀𝐶.

Now we need to consider how to find the measure of this angle. So let’s look at the other information given in the figure. We know that the measure of angle 𝐴𝐵𝑌 is 51.8 degrees, and we’re told that the line segments 𝐵𝐴 and 𝑀𝐶 are parallel. Angles 𝐵𝑀𝐶 and 𝑀𝐵𝐴 are therefore alternate angles in parallel lines and so are of equal measure.

The final detail to observe is that the angle 𝑀𝐵𝑌 is the angle formed between the radius 𝑀𝐵 and the tangent 𝑋𝑌 at the point of tangency. We can then recall a second key theorem, which is that a tangent to a circle is perpendicular to the radius at the point of contact. This means that the measure of angle 𝑀𝐵𝑌 is 90 degrees. We can therefore calculate the measure of angle 𝑀𝐵𝐴 by subtracting 51.8 degrees from 90 degrees, giving 38.2 degrees. The measure of angle 𝐵𝑀𝐶 is also 38.2 degrees, due to alternate angles in parallel lines being equal.

We’re finally able to calculate the measure of angle 𝐶𝐵𝑋. It’s one-half of the measure of angle 𝐵𝑀𝐶, so it’s one-half of 38.2 degrees. That’s 19.1 degrees. So, by identifying that angle 𝐶𝐵𝑋 is an angle of tangency and then recalling that the measure of an angle of tangency is half the measure of the central angle subtended by the same arc, along with using other key angle properties, we’ve found that the measure of angle 𝐶𝐵𝑋 is 19.1 degrees.