### Video Transcript

Given that line π΄π΅ is a tangent to a circle with center π and the measure of angle
π΄π΅π is equal to 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 in 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 a measure of 49 degrees.

From the information given, we arenβt able to calculate the measure of 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 is that line π΄π΅ is a
tangent to the circle with center π. And a 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 measure of angle π΅π΄π is 90 degrees. Now we know one more angle within the figure. However, we still arenβt able to calculate the measure of angle π΄π·π΅ directly, so
letβs see what other angles we can work out.

We have a triangle. In fact, triangle π΄ππ΅ is a right triangle, and we know two of its angles, the
right angle and the angle measuring 49 degrees. So, using the fact that the angle measures in a triangle sum to 180 degrees, we can
calculate the third angle in this triangle. We do this by writing an equation which states that the measure of angle π΄ππ΅ plus
49 degrees plus 90 degrees is equal to 180 degrees. 49 plus 90 is 139, and subtracting this from 180, we find the measure of angle π΄ππ΅
is 41 degrees. So we now know another angle in our diagram.

We still donβt have enough information to calculate the measure of an angle π΄π·π΅,
but we can now calculate a different angle, angle π΄ππ·. We know that the angle measures on a straight line sum to 180 degrees. So, the measure of angle π΄ππ· plus the angle measure weβve just calculated, 41
degrees, must equal 180 degrees. The measure of angle π΄ππ· is therefore equal to 180 degrees minus 41 degrees. Thatβs 139 degrees. Now we found almost all the angles in the figure, but still not the one we were
looking for.

The final step is to consider triangle π΄ππ·, in which we know one angle is 139
degrees. We recognize both line segment ππ΄ and line segment ππ· as radii of the same circle
with center π. Therefore, they have the same length. It follows that triangle ππ·π΄ is an isosceles triangle, and this means that angles
π·π΄π and π΄π·π have equal measure. We can therefore find the measure of each angle by subtracting the measure of the
third angle, 139 degrees, from the total angle sum in a triangle, 180 degrees, and
then splitting the remainder in half. Doing so gives each of these angles a measure of 20.5 degrees.

Now we know that angle π΄π·π is in fact the same as angle π΄π·π΅. They both refer to the angle highlighted here in pink. And so, weβve completed the problem. By using some of the more basic facts of angles in triangles and angles in straight
lines and the key property that the tangent to a circle is perpendicular to the
radius at the point of contact, we found that the measure of angle π΄π·π΅ is 20.5
degrees.