### Video Transcript

Given that the measure of angle
π΅πΈπΆ is 31 degrees, find the measure of angle πΆ and the measure of angle
π΅π·π΄.

Weβve been given that angle π΅πΈπΆ
is 31 degrees. Thatβs our first piece of
information. We wanna go ahead and list out the
other things we know based on the diagram. We can say that line segment ππΉ,
line segment ππ΄, line segment ππΈ, and line segment ππ΅ are all radii of the
circle π. And we can say that ray πΆπ΅ is
tangent to this circle at the point π΅. We want to know the measure of
angle πΆ, which is here, and the measure of angle π΅π·π΄, which is here.

We wanna use our given statements
and then draw some conclusions. First of all, we can say that all
of the radii will be equal in length to one another because we know the definition
of a radius. We can also say that triangle
π΄ππΉ and triangle ππΈπ΅ are isosceles triangles because they both have two sides
of the triangle that are equal in length. We can also say that the measure of
angle π΄ππΉ will be equal to the measure of angle π΅ππΈ because theyβre vertical
angles. And that means we can say that
triangle π΅ππΈ is congruent to triangle π΄ππΉ because they have a side, an angle,
and a side that are congruent. And since these triangles are
congruent, all of these angles will be equal to one another. They will be congruent angles, all
measuring in at 31 degrees.

Something else we can now identify
is that the measure of angle π΄π΅πΆ is 90 degrees because of tangent line
properties. So far, weβve identified a lot of
the angles, but not the ones we need to find. We now wanna focus in on the
triangle created from points πΈ, πΆ, and π΅. The angle πΆπ΅πΈ is created from a
90-degree angle and a 31-degree angle. That means the angle πΆπ΅πΈ is 121
degrees. The measure of angle πΆ plus 121
degrees plus 31 degrees must equal 180 degrees. 121 plus 31 equals 152. We subtract that value from both
sides of the equation. And we get that the measure of
angle πΆ is 28 degrees.

To find the measure of angle
π΅π·π΄, we need to follow a really similar procedure. Weβre gonna focus on the triangle
π΄π΅π·, which has a right angle and a 31-degree angle. In this case, the measure of angle
π΅π·π΄ plus 90 degrees plus 31 degrees will equal 180 degrees. 90 plus 31 is 121. So we subtract 121 from both sides
of the equation, which tells us that the measure of angle π΅π·π΄ is 59 degrees. And we found both of these answers
by knowing that the sum of the interior angles in a triangle must be equal to 180
degrees.