### Video Transcript

Is the trapezoid π΄π΅πΆπ· a cyclic quadrilateral?

Letβs start by reminding ourselves that a cyclic quadrilateral is a quadrilateral whose vertices are inscribed on a circle. There are a few different angle properties that we can use to prove if a quadrilateral is cyclic or not.

Given that we have the diagonals marked, we may choose to 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. For example, if we could show that the angle π΄π·π΅ was equal in measure to angle π΄πΆπ΅, or indeed if we had another angle pair, for example, angle π·π΄πΆ equal to angle π·π΅πΆ, then the quadrilateral would be cyclic.

So letβs see if we can work out the angle measure of π΅πΆπ΄. Because weβre given that π΄π΅πΆπ· is a trapezoid, that means thereβll be one set of parallel sides. And we can see them here. Sides π΅πΆ and π΄π· are parallel. Therefore, angle π·π΅πΆ is alternate to this given angle of π΄π·π΅. And so theyβre both equal to 84 degrees.

Using the fact that the interior angles in a triangle add up to 180 degrees will allow us to find our unknown angle. We can write that the three angle measures of 84 degrees plus 52 degrees plus the measure of angle π΅πΆπ΄ is equal to 180 degrees. We can simplify this and then subtract 136 degrees from both sides, giving us that the measure of angle π΅πΆπ΄ is 44 degrees.

We can now really easily see that this angle measure of π΅πΆπ΄ is not equal to the angle measure at π΅π·π΄. Therefore, we can give the answer no, trapezoid π΄π΅πΆπ· is not a cyclic quadrilateral.