### Video Transcript

Given that the measure of angle
π΅π΄π· is π₯ plus 34 degrees, find the values of π₯ and π¦.

Weβre told in the question that
angle π΅π΄π· is equal to π₯ plus 34 degrees. We also notice from the diagram
that triangle π΅π΄πΈ is isosceles, as the lengths π΅π΄ and π΅πΈ are equal. This means that angle π΅π΄πΈ is
equal to angle π΅πΈπ΄, which is equal to 51 degrees. We know that the angles in a
triangle sum to 180 degrees. This means that the angle π΄π΅πΈ
plus 51 degrees plus 51 degrees is equal to 180 degrees. 51 plus 51 is equal to 102. And subtracting this from 180 gives
us angle π΄π΅πΈ is 78 degrees.

We also know that angles on a
straight line sum to 180 degrees. This means that we can calculate
the angle π΄π΅πΆ inside the cyclic quadrilateral by subtracting 78 degrees from 180
degrees. Angle π΄π΅πΆ is equal to 102
degrees. The quadrilateral π΄π΅πΆπ· is
cyclic as all four vertices lie on the circumference of a circle. We know that opposite angles in any
cyclic quadrilateral sum to 180 degrees. This means that π₯ plus 102 is
equal to 180 and π¦ plus π₯ plus 34 is also equal to 180.

As just mentioned by looking at the
angles at vertices π΅ and π·, we have the equation π₯ plus 102 degrees is equal to
180 degrees. Subtracting 102 from both sides of
this equation gives us π₯ is equal to 78 degrees. The angles at vertices π΄ and πΆ
will also sum to 180 degrees. 78 plus 34 is equal to 112. Subtracting this from both sides of
the equation gives us π¦ is equal to 68 degrees. The values of π₯ and π¦,
respectively, are 78 degrees and 68 degrees. We could check this by adding the
four angles inside the quadrilateral, which must sum to 360 degrees.