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

Given that π‘šβˆ π΅π΄π· = (π‘₯ + 34)Β°, find the values of π‘₯ and 𝑦.


Video Transcript

Given that the measure of angle 𝐡𝐴𝐷 is π‘₯ plus 34 degrees, find the values of π‘₯ and 𝑦.

We’re told in the question that angle 𝐡𝐴𝐷 is equal to π‘₯ plus 34 degrees. We also notice from the diagram that triangle 𝐡𝐴𝐸 is isosceles, as the lengths 𝐡𝐴 and 𝐡𝐸 are equal. This means that angle 𝐡𝐴𝐸 is equal to angle 𝐡𝐸𝐴, which is equal to 51 degrees. We know that the angles in a triangle sum to 180 degrees. This means that the angle 𝐴𝐡𝐸 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 𝐴𝐡𝐸 is 78 degrees.

We also know that angles on a straight line sum to 180 degrees. This means that we can calculate the angle 𝐴𝐡𝐢 inside the cyclic quadrilateral by subtracting 78 degrees from 180 degrees. Angle 𝐴𝐡𝐢 is equal to 102 degrees. The quadrilateral 𝐴𝐡𝐢𝐷 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 𝑦 plus π‘₯ plus 34 is also equal to 180.

As just mentioned by looking at the angles at vertices 𝐡 and 𝐷, 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 𝐴 and 𝐢 will also sum to 180 degrees. 78 plus 34 is equal to 112. Subtracting this from both sides of the equation gives us 𝑦 is equal to 68 degrees. The values of π‘₯ and 𝑦, 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.

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