Given that the measure of angle
𝐵𝐴𝐷 is 𝑥 plus 34 degrees, find the values of 𝑥 and 𝑦.
We’re told in the question that
angle 𝐵𝐴𝐷 is equal to 𝑥 plus 34 degrees. We also notice from the diagram
that triangle 𝐵𝐴𝐸 is isosceles, as the lengths 𝐵𝐴 and 𝐵𝐸 are equal. This means that angle 𝐵𝐴𝐸 is
equal to angle 𝐵𝐸𝐴, which is equal to 51 degrees. We know that the angles in a
triangle sum to 180 degrees. This means that the angle 𝐴𝐵𝐸
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 𝐴𝐵𝐸 is 78 degrees.
We also know that angles on a
straight line sum to 180 degrees. This means that we can calculate
the angle 𝐴𝐵𝐶 inside the cyclic quadrilateral by subtracting 78 degrees from 180
degrees. Angle 𝐴𝐵𝐶 is equal to 102
degrees. The quadrilateral 𝐴𝐵𝐶𝐷 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 𝑥 plus 102 is
equal to 180 and 𝑦 plus 𝑥 plus 34 is also equal to 180.
As just mentioned by looking at the
angles at vertices 𝐵 and 𝐷, we have the equation 𝑥 plus 102 degrees is equal to
180 degrees. Subtracting 102 from both sides of
this equation gives us 𝑥 is equal to 78 degrees. The angles at vertices 𝐴 and 𝐶
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 𝑥 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.