Find the values of 𝑥 and 𝑦.
In the figure below, we’re given a quadrilateral 𝐴𝐵𝐶𝐷 and we note that all four vertices of this quadrilateral are inscribed on the circle. This means that 𝐴𝐵𝐶𝐷 is a cyclic quadrilateral. An important property which we might then be able to apply is that opposite angles in a cyclic quadrilateral are supplementary. They add to 180 degrees. The first thing we might observe then is that the angle at 𝐴 is opposite to the angle at 𝐶. We could therefore write the equation that four 𝑥 degrees plus 68 degrees is equal to 180 degrees. Do be careful because a very common mistake when we’re working with cyclic quadrilaterals is to forget that opposite angles are supplementary and mistakenly think that opposite angles are equal.
So because four 𝑥 plus 68 degrees equals 180 degrees, then four 𝑥 degrees must be equal to 180 degrees subtract 68 degrees. And that’s 112 degrees. When we divide through by four, we get that 𝑥 degrees is equal to 28 degrees, or more simply 𝑥 is equal to 28. Now that we have found 𝑥, let’s see how we can find 𝑦. The angle which is opposite to the angle at 𝐵, which is 𝑦 degrees, is the angle at 𝐷. But we’re not given a measurement for it. Even if we worked out that the measurement of angle 𝐴 is 112 degrees, that isn’t enough to allow us to find the measurement of angle 𝐵.
So let’s use the other piece of information that we’re given in the diagram. And that is that these line segments of 𝐴𝐵 and 𝐴𝐷 are congruent. Both of these line segments are in fact chords of the circle. Let’s imagine that we create this triangle 𝐴𝐵𝑀 and another triangle here at 𝐴𝐷𝑀. 𝐵𝑀 and 𝐴𝑀 are both radii of the circle, and so they’ll be the same length. 𝐷𝑀 is also a radius, and so it will be congruent with the other two radii. So what can we say about this triangle 𝐴𝐵𝑀? Well, we know that it’s going to be an isosceles triangle because it’s got two sides that are the same length. And therefore, if angle 𝐴𝐵𝑀 is 𝑦 degrees, then angle 𝐵𝐴𝑀 must also be 𝑦 degrees.
In the same way, we can also say that this triangle 𝐴𝑀𝐷 is also an isosceles triangle. More importantly, however, we can also say that triangle 𝐴𝐵𝑀 is congruent to triangle 𝐴𝐷𝑀. We can say this by applying the SSS congruency criterion. There are three corresponding sides which are congruent. And therefore, we can say that the measure of angle 𝑀𝐴𝐷 must also be 𝑦 degrees and so must the measure of angle 𝑀𝐷𝐴. We were given that this angle 𝐵𝐴𝐷 is four 𝑥 degrees, and we know that four 𝑥 degrees is 112 degrees. Because that’s equal to two 𝑦, then we can divide through by two to find that 𝑦 degrees is equal to 56 degrees. We can then give the answer that 𝑥 is equal to 28 and 𝑦 is equal to 56.