Video Transcript
Is 𝐴𝐵𝐶𝐷 a cyclic quadrilateral?
A cyclic quadrilateral is a quadrilateral whose vertices are inscribed on a circle. One way in which we can prove a quadrilateral is cyclic is by checking the angles made with the diagonals. For example, if we could demonstrate that the measure of angle 𝐷𝐴𝐶 was equal to the measure of angle 𝐷𝐵𝐶, then that would show that 𝐴𝐵𝐶𝐷 was cyclic. Another pair of angles we could check would be the measure of angle 𝐴𝐷𝐵 and the measure of angle 𝐴𝐶𝐵. Showing that these two angle measures are equal would show that the quadrilateral was cyclic. But let’s look at the first pair of angles. This will mean that we need to work out the angle measure of angle 𝐷𝐵𝐶.
We have a triangle here 𝐵𝐶𝐸, which will help us, but of course we need two angles in order to find the missing one. Using the fact that the angles on a straight line sum to 180 degrees will allow us to work out this angle measure of 𝐶𝐸𝐵. It will be 180 degrees subtract 83 degrees, which leaves us with 97 degrees. Now, we have the two angles in the triangle, we can calculate this unknown angle measure of 𝐷𝐵𝐶.
Because we know that the interior angle measures in a triangle add up to 180 degrees, we have 41 degrees plus 97 degrees plus the measure of angle 𝐷𝐵𝐶 is equal to 180 degrees. Simplifying the left-hand side, we have 138 degrees plus the measure of angle 𝐷𝐵𝐶 is 180 degrees. When we subtract 138 degrees from both sides, that leaves us with the measure of angle 𝐷𝐵𝐶 is 42 degrees. This now demonstrates that we have two equal angle measures. The measure of angle 𝐷𝐵𝐶 is equal to the measure of angle 𝐷𝐴𝐶. And so, 𝐴𝐵𝐶𝐷 is a cyclic quadrilateral.
We could also have investigated the other pair of angles. And we can choose this pair of angles at the diagonals because we were given the angle measure of 𝐵𝐶𝐴. Using the straight line 𝐴𝐶, we could have worked out that this angle measure at 𝐷𝐸𝐴 is also 97 degrees. Using the triangle 𝐴𝐸𝐷 and the fact that the angles add up to 180 degrees would have allowed us to show that angle 𝐴𝐷𝐵 is 41 degrees. This would also show that an angle made with a diagonal and side is equal to the angle made with the other diagonal and opposite side. Note that we don’t need to show that both pairs of angles are congruent. Just one of these is sufficient to prove that 𝐴𝐵𝐶𝐷 is cyclic. And so, we can give the answer yes, 𝐴𝐵𝐶𝐷 is a cyclic quadrilateral.
