### Video Transcript

In the figure, point π· is on the line π΄πΆ, π΄π΅ equals π΅π· which equals πΆπ·, and π΄π· equals 18. What is the measure, in degrees, of angle π΅πΆπ·?

Letβs start by labeling this image π΄π΅ equals π΅π· which equals πΆπ·. π΄π΅ is the same length as π΅π·. And itβs the same length as πΆπ·. What can we say then about triangle π΄π΅π·? We can say that triangle π΄π΅π· is isosceles. And in an isosceles triangle, the angles opposite the same length sides have the same measure. And so we can say that angle π΄π·π΅ measures 30 degrees, just like angle π΅π΄π·. What other information were given is that π΄π· measures 18 and point π· is on the line π΄πΆ. Because point π· is on the line π΄πΆ, we know something about the angle πΆπ·π΅. Angle πΆπ·π΅ and angle π΅π·π΄ must measure 180 degrees because they make a straight line.

We know that the measure of πΆπ·π΄ Is 30 degrees. This means the measure of angle πΆπ·π΅ is equal to 180 degrees minus 30 degrees. Angle πΆπ·π΅ measures 150 degrees. If we look closely at the other triangle, triangle π΅π·π΄, it is also an isosceles triangle. It has two sides that are the same length. This tells us that the angle π·π΅πΆ and the angle π·πΆπ΅ must be equal to one another. We could label them as π₯ degrees.

And because we know that inside of a triangle all the angles add up to 180 degrees, we can say 150 degrees plus π₯ plus π₯ must equal 180 degrees. π₯ plus π₯ equals two π₯ and then we subtract 150 degrees from both sides of the equation. Two π₯ equals 30 degrees. And then we divide both sides by two, and we see that π₯ equals 15 degrees. The two other angles in this isosceles triangle must measure 15 degrees each. Weβre interested in angle π΅πΆπ·, which is here. Angle π΅πΆπ· must measure 15 degrees.