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Is π΄π΅πΆπ· a cyclic quadrilateral?

We begin by recalling that there are two ways we can prove that a quadrilateral is cyclic: firstly, if the opposite angles in the quadrilateral sum to 180 degrees and secondly if an exterior angle is equal to the interior angle at the opposite vertex. It is the first of these we will use in this question. If we can prove that the measure of angle π΅ plus the measure of angle π· is equal to 180 degrees, then the quadrilateral is cyclic. We can also do this with angles π΄ and πΆ. We begin by noticing that triangle π΄π·πΆ is isosceles. This means that the measure of angle πΆπ΄π· is equal to the measure of angle π΄πΆπ·, which is equal to 53 degrees. Since angles in a triangle sum to 180 degrees, the measure of angle π΄π·πΆ is equal to 180 degrees minus 53 degrees plus 53 degrees. This is therefore equal to 74 degrees.

We now have the measures of two opposite angles in our quadrilateral. 106 plus 74 is equal to 180. So the measure of angle π΅ and the measure of angle π· do sum to 180 degrees. And we can therefore conclude that π΄π΅πΆπ· is a cyclic quadrilateral, and the correct answer is yes.

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