# Video: Finding the Measures of the Interior Angles of a Cyclic Quadrilateral given the Measure of One Angle in It

Given that πβ π΅π΄π· = (π₯ + 34)Β°, find the values of π₯ and π¦.

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### Video Transcript

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

Weβre 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΄πΈ is isosceles, as the lengths π΅π΄ and π΅πΈ are equal. This means that angle π΅π΄πΈ is equal to angle π΅πΈπ΄, 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΅πΈ 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΅πΈ 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΅πΆ inside the cyclic quadrilateral by subtracting 78 degrees from 180 degrees. Angle π΄π΅πΆ is equal to 102 degrees. The quadrilateral π΄π΅πΆπ· 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 πΆ 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.