Question Video: Finding the Measure of an Angle given the Measures of its Minor Arc and Major Arc Mathematics • Third Year of Preparatory School

Video Transcript

Find 𝑥.

Let’s have a look at the diagram we’ve been given. We can see that we have a circle. And 𝑥 degrees is the measure of the angle formed by the intersection of two chords. Those are the line segments 𝐵𝐸 and 𝐶𝐷. The angles of intersecting chords theorem tells us that the measure of the angle formed by the intersection of two chords inside a circle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. The arc intercepted by the angle marked 𝑥 degrees is the arc 𝐶𝐸, and the arc intercepted by its vertical angle is the arc 𝐵𝐷. So we can form an equation. 𝑥 degrees is equal to one-half the measure of the arc 𝐶𝐸 plus the measure of the arc 𝐵𝐷.

Now, we can’t solve this equation because we don’t know the measures of either of these two arcs. So let’s consider the other information given in the diagram. There are also two secant segments of the circle, the line segments 𝐴𝐶 and 𝐴𝐸, which intersect at a point outside the circle. And we’re given the measure of the angle between them. The angles of intersecting secants theorem tells us that the measure of the angle formed by two secants that intersect outside a circle is half the positive difference of the measures of the two intercepted arcs. The intercepted arcs are the arcs 𝐵𝐷 and 𝐶𝐸. 𝐵𝐷 connects the two points where each secant first intersects the circle. And 𝐶𝐸 connects the two points where they each intercept the circle for a second time. From the figure, we can see that the arc 𝐶𝐸 has the larger measure. And so we can form a second equation. 40 degrees is equal to one-half the measure of the arc 𝐶𝐸 minus the measure of the arc 𝐵𝐷.

We now have two simultaneous equations involving the measure of the arc 𝐶𝐸 and the measure of the arc 𝐵𝐷. But we currently have insufficient information to solve them. The final piece of information given in the question which we’ve not yet used is the measure of the inscribed angle, angle 𝐵𝐸𝐷. Recalling that the measure of an inscribed angle is equal to half the measure of its intercepted arc, we have that the measure of angle 𝐵𝐸𝐷 is one-half the measure of the arc 𝐵𝐷.

We therefore have enough information to calculate the measure of the arc 𝐵𝐷. Substituting 30 degrees for the measure of angle 𝐵𝐸𝐷, we have that 30 degrees is equal to one-half the measure of the arc 𝐵𝐷. And then multiplying both sides by two, we find that the measure of the arc 𝐵𝐷 is 60 degrees.

We’re now in a position to work backwards through all the working we’ve done to calculate the value of 𝑥. First, we can substitute the value 60 degrees for the measure of the arc 𝐵𝐷 in our second equation. This will enable us to calculate the measure of the arc 𝐶𝐸. And then we can substitute both values for the measure of the arc 𝐶𝐸 and the measure of the arc 𝐵𝐷 into our first equation to calculate 𝑥. So, substituting 60 degrees for the measure of the arc 𝐵𝐷 in our second equation, we have that 40 degrees is equal to one-half the measure of the arc 𝐶𝐸 minus 60 degrees. We can then multiply both sides of this equation by two to eliminate the fraction. And it gives 80 degrees is equal to the measure of the arc 𝐶𝐸 minus 60 degrees. Finally, we add 60 degrees to each side of the equation. And we find that the measure of the arc 𝐶𝐸 is 140 degrees.

Lastly then, we return to our very first equation and we substitute 140 degrees for the measure of the arc 𝐶𝐸 and 60 degrees for the measure of the arc 𝐵𝐷. We have that 𝑥 degrees is one-half of 140 degrees plus 60 degrees. That’s one-half of 200 degrees, which is 100 degrees. Now, 𝑥 is purely the numeric part of this answer, so the value of 𝑥 is 100. So, by recalling the theorems for the angles between intersecting chords and the angles between intersecting secants, we found that the value of 𝑥 is 100.