Find the measure of angle 𝐵𝐶𝐷.
So we’ve been given a diagram of a circle with centre 𝑀 and a quadrilateral 𝐴𝐵𝐶𝐷. The angle whose measure we’ve been asked to find is the angle formed when we travel from 𝐵 to 𝐶 to 𝐷. So it’s this angle that I’ve marked here in orange. We can see that we’ve been given two other angles in the diagram. They’re the two angles in the triangle 𝐴𝐵𝐶. There’s a right angle and an angle of 64 degrees. This means that straightaway, we could calculate the third angle in this triangle, the angle marked in green, as the angle sum in a triangle is always 180 degrees. This will tell us what part of the angle 𝐵𝐶𝐷 is.
So we have that the measure of angle 𝐵𝐶𝐴 is equal to 180 degrees minus 90 degrees minus 64 degrees. It’s 26 degrees. Remember, we’re looking to calculate the angle 𝐵𝐶𝐷. We now know that part of it is 26 degrees. But we need to work out what the other part is, the angle 𝐴𝐶𝐷. Let’s think about how to do this. If we consider the quadrilateral 𝐴𝐵𝐶𝐷, we can see that it is in fact a cyclic quadrilateral because all four of its vertices lie on the circumference of the circle. A key fact about cyclic quadrilaterals, which isn’t true of quadrilaterals in general, is that their opposite angles sum to 180 degrees.
We’ve been given one of the angles in this cyclic quadrilateral. Angle 𝐴𝐵𝐶 is 64 degrees. Therefore, we can calculate the opposite angle, angle 𝐴𝐷𝐶. Using the reasoning just described, it’s equal to 180 degrees minus 64 degrees, which is 116 degrees. So we’re getting closer to finding the angle that we were originally asked to calculate. Now let’s think about the triangle 𝐴𝐷𝐶. A key piece of information given in the question, that we haven’t used yet, is that two of the sides in this triangle are equal in length. These blue lines on the sides 𝐴𝐷 and 𝐶𝐷 indicate that they’re congruent. And therefore, the triangle 𝐴𝐷𝐶 is an isosceles triangle.
The two currently unknown angles in this triangle are the base angles. And therefore, they’re equal to one another. This means they can each be calculated by finding half of the remaining angle sum in this triangle. 180 degrees minus the known angle of 116 degrees and then divided by two, for the two base angles. Each of the base angles are equal to 32 degrees. So I’ve added that information to the diagram.
Remember, our overall objective in this question was to find the measure of the angle 𝐵𝐶𝐷, which we can now see is made up of an angle of 26 degrees and an angle of 32 degrees. So to find the sum of this angle, we need to add those two parts together. The measure of angle 𝐵𝐶𝐷 is 58 degrees.
Remember, the key fact that we used in this question, other than facts about the angle sum in a triangle, is that if a quadrilateral is cyclic, meaning all four of its vertices lie on the circumference of a circle, then the opposite angles must sum to 180 degrees.