Find the value of 𝑧.
Let’s look at the diagram that we’ve been given. It shows a circle and then two lines 𝐴𝐶 and 𝐷𝐶. These lines are both secants as they each intersect the circle in two places. We’re asked to find the value of 𝑧, which we can see is the measure of the arc 𝐵𝐸.
We’ve been told the measure of the arc 𝐴𝐷 and the measure of the angle where the two secants meet. In order to answer this question, we need to recall the relationship that exists between the measures of these two arcs and the angle. The relationship is this: If two secants intersect in the exterior of a circle, the measure of the angle formed is half the difference of the measures of the intercepted arcs.
The intercepted arcs here are arcs 𝐴𝐷 and 𝐵𝐸, with 𝐴𝐷 being the longer arc. So we have that the measure of angle 𝐶 is half the difference between the measures of the two arcs, half of the measure of arc 𝐴𝐷 minus the measure of arc 𝐵𝐸.
We can substitute 125 degrees, 𝑧 degrees, and 35 degrees into the relevant places in this equation. We have that 35 degrees is equal to a half of 125 degrees minus 𝑧 degrees. This is an equation that we can solve in order to find the value of 𝑧. The first step is to multiply both sides of the equation by two. This gives 70 degrees is equal to 125 degrees minus 𝑧 degrees.
Now we want to find the value of 𝑧. And it currently has a negative coefficient. So I’m going to add 𝑧 degrees to both sides of the equation. This gives 𝑧 degrees plus 70 degrees is equal to 125 degrees. To find the value of 𝑧, we need to subtract 70 degrees from both sides. We have that 𝑧 degrees is equal to 125 degrees minus 70 degrees, which is 55 degrees. The value of 𝑧 is 55.
Remember, the key property we used in this question is that if two secants intersect in the exterior of a circle, the measure of the angle formed is half the difference of the measures of the intercepted arcs.