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.
