### Video Transcript

Given that the line π΄π΅ is a tangent to the circle with center π and the measure of
angle ππ΅πΉ is 123 degrees, determine the measure of angle π΄ππ΅.

Angle π΄ππ΅ is the angle formed when we travel from π΄ to π and then to π΅. Angle ππ΅πΉ is the angle made when we travel from π to π΅ to πΉ, and weβre told
that its measure is 123 degrees. We can see that the angle weβre looking to find, π΄ππ΅, is contained within a
triangle. If we can work out the other two angles in this triangle, we can use the fact that
the sum of the angles in any triangle is 180 degrees to find the angle weβre looking
for.

First, letβs consider the angle ππ΅π΄. One of our most basic angle facts is that the angles on a straight line sum to 180
degrees. And this angle is on a straight line with the angle weβve already marked as being 123
degrees. So we can say that the measures of angles ππ΅πΉ and ππ΅π΄ sum to 180 degrees. We can substitute 123 degrees for the measure of angle ππ΅πΉ into our equation. And we now have an equation we can solve to find the measure of angle ππ΅π΄. To do this, we need to subtract 123 degrees from both sides of the equation. And doing so gives us that the measure of angle ππ΅π΄ is equal to 57 degrees. So weβve found one of the angles in triangle ππ΅π΄. Can we find another one? What about angle ππ΄π΅?

Well, this is the angle formed where a tangent to the circle, thatβs the line π΄π΅,
meets the radius of the circle, π΄π. And we know that a tangent to a circle is perpendicular to the radius at the point of
contact. So we know that angle ππ΄π΅ is a right angle, so it has measure 90 degrees.

Weβve therefore found two of the angles in triangle ππ΅π΄. And using the angle sum in a triangle, we can find the third. Since the angles in a triangle sum to 180 degrees, we have the equation the measure
of angle π΄ππ΅ plus 90 degrees plus 57 degrees equals 180 degrees. Subtracting 90 degrees and 57 degrees from both sides then, we have 180 minus 90
minus 57 degrees on our right-hand side. And this is equal to 33 degrees. And so the measure of angle π΄ππ΅ equals 33 degrees.