### Video Transcript

Determine the measure of the arc πΆπ΅.

Letβs have a closer look at the diagram weβve been given. We have a circle and the line segments π΄πΆ and π΄πΈ. Now, these are each segments of secants of the circle because they each intersect the circle in two places and pass outside the circle. These two secant segments intersect at a point outside the circle, point π΄. And weβve been given the measure of the angle formed between them.

The other information weβre given is the measure of the arc πΆπΈ, which is the larger of the two arcs intercepted by these two secant segments. Now, weβre asked to determine the measure of the arc πΆπ΅. The only thing we know about this arc is that it is the same length as the arc πΈπ·. Letβs think about how we can use the fact that we know the angle formed between these two secant segments to help us.

Well, the intersecting secant theorem tells us that the angle between two secants or secant segments that intersect outside a circle is half the positive difference of the measures of the arcs intercepted by the sides of the angle. The arcs intercepted by the sides of the angle at π΄ are π΅π· and πΆπΈ. πΆπΈ is clearly the larger of these two arcs, so we can form an equation. 34 degrees is equal to one-half the measure of the arc πΆπΈ minus the measure of the arc π΅π·. Now, we know the measure of the arc πΆπΈ; itβs 151 degrees. So we could substitute this value into the equation and then solve it to find the measure of the arc π΅π·. But how will this help us?

Well, we know that the measure of the entire circumference of a circle is 360 degrees. So the measure of the arc πΆπΈ plus the measure of the arc π΅π· plus the measure of the arc πΆπ΅ plus the measure of the arc π·πΈ must be 360 degrees. We know the measure of the arc πΆπΈ. Weβve just discussed how we can find the measure of the arc π΅π·. And itβs the measure of the arc πΆπ΅ we want to find. The measure of the arc π·πΈ, remember, is the same as the measure of the arc πΆπ΅. So in fact, we have one less unknown than we thought.

We can change our equation to the measure of the arc πΆπΈ plus the measure of the arc π΅π· plus twice the measure of the arc πΆπ΅ is equal to 360 degrees. And now we see that once weβve determined the measure of the arc π΅π·, weβll be able to find the measure of the arc πΆπ΅. Returning to our earlier equation then, we can multiply both sides by two, which will eliminate the fraction on the right-hand side and give 68 on the left-hand side. We can also substitute 151 degrees for the measure of the arc πΆπΈ. And we have 68 degrees equals 151 degrees minus the measure of the arc π΅π·. We can add the measure of the arc π΅π· to each side of this equation and then subtract 68 degrees from each side. And we find that the measure of the arc π΅π· is 83 degrees.

We can now substitute the measures of the arcs πΆπΈ and π΅π· into our second equation. And we have 151 degrees plus 83 degrees plus twice the measure of the arc πΆπ΅ is equal to 360 degrees. Subtracting 151 and 83 degrees from each side of this equation, we find that twice the measure of the arc πΆπ΅ is 126 degrees. Finally, we can divide both sides of this equation by two to give the measure of the arc πΆπ΅ is 63 degrees. So, by recalling the angles between intersecting secants theorem and the fact that the measure of the entire circumference of a circle is 360 degrees, we found that the measure of the arc πΆπ΅ is 63 degrees.