### Video Transcript

Suppose that the line ππ is a
tangent to the circle with center π at π΅, line segment π΅π΄ is parallel to line
segment ππΆ, and the measure of angle π΄π΅π is 51.8 degrees. Find the measure of angle
πΆπ΅π.

Letβs begin by marking the angle
weβve been given and the angle we wish to calculate on the diagram. Angle π΄π΅π is 51.8 degrees, and
weβre looking for the measure of angle πΆπ΅π. We can now see that the angle whose
measure weβve been asked to calculate is an angle of tangency, as it is the angle
between the tangent ππ and the chord π΅πΆ. 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 π΅πΆ is the minor arc π΅πΆ, and the central angle subtended by this arc is
angle π΅ππΆ. So we have that the measure of
angle πΆπ΅π is half the measure of angle π΅ππΆ.

Now we need to consider how to find
the measure of this angle. So letβs look at the other
information given in the figure. We know that the measure of angle
π΄π΅π is 51.8 degrees, and weβre told that the line segments π΅π΄ and ππΆ are
parallel. Angles π΅ππΆ and ππ΅π΄ are
therefore alternate angles in parallel lines and so are of equal measure.

The final detail to observe is that
the angle ππ΅π is the angle formed between the radius ππ΅ and the tangent ππ 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 ππ΅π is 90 degrees. We can therefore calculate the
measure of angle ππ΅π΄ by subtracting 51.8 degrees from 90 degrees, giving 38.2
degrees. The measure of angle π΅ππΆ is also
38.2 degrees, due to alternate angles in parallel lines being equal.

Weβre finally able to calculate the
measure of angle πΆπ΅π. Itβs one-half of the measure of
angle π΅ππΆ, so itβs one-half of 38.2 degrees. Thatβs 19.1 degrees. So, by identifying that angle
πΆπ΅π 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, weβve found that the measure of angle πΆπ΅π
is 19.1 degrees.