### Video Transcript

Given that the line segment π΄π΅ is
a tangent to the circle π, and the measure of angle π΄π΅π is 49 degrees, determine
the measure of angle π΄π·π΅.

Angle π΄π·π΅ is the angle formed
when we travel from π΄ to π· to π΅. So, itβs the angle marked in orange
on the diagram. Angle π΄π΅π is the angle formed
when we travel from π΄ to π΅ to π. Itβs the angle now marked in pink
on the diagram with its measure of 49 degrees. From the information given, we
arenβt able to calculate angle π΄π·π΅ directly. Weβre going to need to find the
measures of some other angles in the figure first. The other key piece of information
given in the question, though, is that the line π΄π΅ is a tangent to the circle
π. And the key property about tangents
of circles is that a tangent to a circle is perpendicular to the radius at the point
of contact.

The point where the tangent meets
the circle is point π΄. And the radius here is the line
segment π΄π. So, we know that the angle π΅π΄π
is 90 degrees. So, we now know one more angle
within the figure. We still arenβt able to calculate
angle π΄π·π΅ directly. So, letβs see what other angles we
could work out. We have a triangle. In fact, itβs a right triangle,
triangle π΄ππ΅. And we know two of its angles, the
right angle and the angle of 49 degrees. So, using the fact that angles in a
triangle sum to 180 degrees, we can calculate the third angle in this triangle.

We have that angle π΄ππ΅ plus 90
degrees plus 49 degrees equals 180 degrees. 90 plus 49 is 139. And subtracting this from 180, we
find that angle π΄ππ΅ is 41 degrees. So, we now know another angle in
our diagram. We still donβt have enough
information to calculate angle π΄π·π΅. But we can now calculate a
different angle, angle π΄ππ·. We know that the angles on any
straight line sum to 180 degrees. So, angle π΄ππ· plus the angle
weβve just calculated of 41 degrees must equal 180 degrees. Angle π΄ππ· is, therefore, equal
to 180 degrees minus 41 degrees. Thatβs 139 degrees.

Now, we found almost all of the
angles in the figure, but still not the one that we were looking for. The final step is to consider
triangle π΄ππ·, in which we know one angle of 139 degrees. We need to notice that both ππ·
and ππ΄ are radii of the circle π. And therefore, theyβre of the same
length. This means that triangle ππ·π΄ is
an isosceles triangle. And it also means that angle ππ·π΄
will be equal to angle ππ΄π·. We can, therefore, find the measure
of each angle by subtracting the third angle, 139 degrees, from the total angle sum
in a triangle, 180 degrees, and then halving the remainder. Doing so gives each of these angles
to be 20.5 degrees. Now, angle ππ·π΄ is in fact the
same angle as angle π΄π·π΅. They both refer to this angle
here. And so, weβve completed the
problem.

By using some of the more basic
facts about angles in triangles and angles in straight lines and then the key
property that a tangent to a circle is perpendicular to the radius at the point of
contact, weβve found the measure of angle π΄π·π΅ is 20.5 degrees.