Video Transcript
Is 𝐴𝐵𝐶𝐷 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 𝐵 plus the measure of angle 𝐷 is equal to 180 degrees, then
the quadrilateral is cyclic. We can also do this with angles
𝐴 and 𝐶. We begin by noticing that
triangle 𝐴𝐷𝐶 is isosceles. This means that the measure of
angle 𝐶𝐴𝐷 is equal to the measure of angle 𝐴𝐶𝐷, which is equal to 53
degrees. Since angles in a triangle sum
to 180 degrees, the measure of angle 𝐴𝐷𝐶 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 𝐵 and
the measure of angle 𝐷 do sum to 180 degrees. And we can therefore conclude
that 𝐴𝐵𝐶𝐷 is a cyclic quadrilateral, and the correct answer is yes.
