### Video Transcript

Find the measure of angle
πΆπ΄π΅.

Letβs begin by marking the angle
whose measure weβve been asked to calculate on the diagram. We can then see that this is an
angle of tangency, as it is the angle between the tangent π΄π΅ and the chord
π΄πΆ. We can therefore recall the
following theorem. The measure of an angle of tangency
is equal to half the measure of the central angle subtended by the same arc. The arc that connects the endpoints
of the chord π΄πΆ is the minor arc π΄πΆ. And if we sketch in the radii
connecting points π΄ and πΆ to the center of the circle, then the central angle
subtended by this arc is the angle marked in pink, angle π΄ππΆ. So we have that the measure of
angle πΆπ΄π΅ is one-half the measure of angle π΄ππΆ.

Now we need to consider how to find
the measure of this angle. And to do so, we need to observe
that each side of triangle π΄πΆπ· is the same length. So it is an equilateral
triangle. If we then sketch in the radius
connecting point π· to the center of the circle, we can conclude that the three
triangles into which we have divided the larger triangle π΄πΆπ· are congruent. This is because each triangle has
two sides that are radii of the circle and one side from the original triangle.

The three triangles therefore have
three side lengths the same as each other and so are congruent by the side-side-side
congruency condition. This also means that the angles in
each triangle that are between the two sides that are radii of the circle are all
congruent. As angles around a point sum to 360
degrees, each of these angles is one-third of 360 degrees, which is 120 degrees. We therefore know that, in
particular, the measure of angle π΄ππΆ is 120 degrees. The measure of angle πΆπ΄π΅ is
therefore one-half of 120 degrees, which is 60 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, weβve
found that the measure of angle πΆπ΄π΅ is 60 degrees.