Video Transcript
In a cyclic quadrilateral 𝐴𝐵𝐶𝐷, if the measure of angle 𝐴 equals three times the
measure of angle 𝐵 equals two times the measure of angle 𝐶, find the measure of
angle 𝐷.
Let’s begin this question by recalling what a cyclic quadrilateral is. A cyclic quadrilateral is a four-sided polygon whose vertices are inscribed on a
circle. We’ve been given that the measure of angle 𝐴 is three times the measure of angle 𝐵,
which is two times the measure of angle 𝐶. So, out of these three angles, angle 𝐴 is the largest angle followed by angle 𝐶 and
then angle 𝐵.
We are asked to find the measure of angle 𝐷. So let’s draw a sketch of this cyclic quadrilateral to help us visualize the
problem. We haven’t been given any angles that we can mark on this diagram. But let’s try to write an expression for each angle instead. 𝐴 is the largest angle. It’s three times the measure of angle 𝐵 and two times the measure of angle 𝐶. So let’s say that the measure of angle 𝐴 is six 𝑥, as six is a multiple of both
three and two. Then, angle 𝐵 would have to be two 𝑥 because 𝐴 is three times the measure of angle
𝐵. And angle 𝐶 would have to be three 𝑥 because 𝐴 is two times the measure of angle
𝐶. And let’s call angle 𝐷 𝑦 because we don’t have any information about the measure of
angle 𝐷.
So now in order to help us work out the value of 𝑥 and 𝑦, let’s recall what we know
about the angles in a quadrilateral. We know that the angle measures in a quadrilateral sum to 360 degrees. But this won’t help us find both 𝑥 and 𝑦. So let’s instead remember the important fact that this is a cyclic quadrilateral. And that means that this quadrilateral has some special properties. In particular, we can recall that the measures of the opposite angles in a cyclic
quadrilateral are supplementary. That is, they add up to 180 degrees. Therefore, the measure of angle 𝐴 plus the measure of angle 𝐶 must equal 180
degrees. And the measure of angle 𝐵 plus the measure of angle 𝐷 must also equal 180
degrees.
Now, remember, we wrote expressions for the measures of angles 𝐴, 𝐵, 𝐶, and
𝐷. So let’s substitute in these expressions. We defined the measure of angle 𝐴 as six 𝑥 and the measure of angle 𝐶 as three
𝑥. So we have that six 𝑥 plus three 𝑥 equals 180 degrees. We can simplify this to nine 𝑥 equals 180 degrees. And then dividing both sides by nine, we have that 𝑥 equals 20 degrees.
Returning to the diagram, we defined angle 𝐴 as six 𝑥 degrees. Six times 20 is 120, so angle 𝐴 has a measure of 120 degrees. Angle 𝐶 was defined as three 𝑥, and three times 20 is 60. So angle 𝐶 has a measure of 60 degrees. And as a check, these opposite angle measures of 120 degrees and 60 degrees do indeed
sum to 180 degrees. Angle 𝐵 was defined as two 𝑥, and two times 20 is 40. So this is 40 degrees.
Now, we need to find the measure of angle 𝐷. There are a few ways in which we could do this. But let’s use the equation we have already written. The measure of angle 𝐵 plus the measure of angle 𝐷 equals 180 degrees. We now know that the measure of angle 𝐵 is 40 degrees. And subtracting 40 degrees from both sides, we have that the measure of angle 𝐷 is
140 degrees.
We can check our answer of 140 degrees is correct in a number of ways, for example,
by using the property that the opposite angles of 40 degrees and 140 degrees sum to
180 degrees or indeed that the sum of all the angles in this cyclic quadrilateral is
360 degrees, thus confirming that the measure of angle 𝐷 is 140 degrees.
