### Video Transcript

Given that the measure of angle π΅πΆπ΄ equals 61 degrees and the measure of angle π·π΄π΅ equals 98 degrees, can a circle pass through the points π΄, π΅, πΆ, and π·?

In this problem, weβre given this figure and weβre asked if the circle can pass through the four points. Letβs consider if we draw in the line segment πΆπ·, then we would have created a quadrilateral π΄π΅πΆπ·. If we have a quadrilateral and there is a circle which passes through the four vertices, then that would be a cyclic quadrilateral. So letβs determine if this quadrilateral π΄π΅πΆπ· is a cyclic quadrilateral. We can fill in the given information that the measure of angle π΅πΆπ΄ is 61 degrees and the measure of angle π·π΄π΅ is 98 degrees.

We can use the angle properties in a quadrilateral to help determine if that quadrilateral is cyclic or not. In this case, weβre given the diagonals. This might give us a clue that we can use the property that if an angle created by a diagonal and side is equal in measure to the angle created by the other diagonal and opposite side, then the quadrilateral is cyclic. We can note that this angle π·π΄π΅ is not an angle created with a diagonal. Itβs an angle created by two sides. However, this angle of π΅πΆπ΄ is an angle created by a diagonal and side. The angle created by the other diagonal and opposite side would occur here at angle π΅π·π΄.

If angle π΅π·π΄ is equal in measure to angle π΅πΆπ΄, then that would mean that π΄π΅πΆπ· is cyclic. We donβt know this angle measure, but letβs see if we can work it out. We can take this triangle of π·π΄π΅ and observe that weβre given these two equal line segments, which means that triangle π·π΄π΅ is an isosceles triangle. This means that the two angles at the base will be equal in length. We can even define them both as something like π₯ degrees.

We then use the fact that the interior angle measures in a triangle add up to 180 degrees. Therefore, the three angle measures of π₯ degrees, π₯ degrees, and 98 degrees must add to give 180 degrees. Simplifying this, we have two π₯ degrees equals 180 degrees minus 98 degrees, which is 82 degrees. When we divide through by two, we find that π₯ degrees is 41 degrees. And so, the two angle measures in this isosceles triangle must be 41 degrees.

But remember that we wanted to find this angle measure of π΅π·π΄ because we wanted to check if it was the same as the angle measure of π΅πΆπ΄. And, of course, 61 degrees is not equal to 41 degrees. And so we can say that the angle created by the diagonal and side is not equal to the angle created by the other diagonal and opposite side.

Therefore, we can give the answer no, there would not be a circle passing through the points π΄, π΅, πΆ, and π·.