### Video Transcript

Determine the measure of arc
πΆπ΅.

First, letβs see what we know. We know that the line segment πΆπ΄
and the line segment πΈπ΄ intersect outside the circle at point π΄. Both line segment πΆπ΄ and line
segment πΈπ΄ are secants of the circle. The figure has also told us that
the measure of arc πΆπ΅ is equal to the measure of arc πΈπ·. We can also say that the sum of all
arc measures in the circle must be 360 degrees. Weβre interested in finding the
measure of arc πΆπ΅. But in order to do that, weβll need
to find the measure of arc π΅π·.

Based on what we said initially
about the intersection outside the circle, we can say that the angle made by two
intersecting lines outside the circle is half the positive difference of the
intercepted arcs, which is the intersecting secant angles theorem. The angle created by these
intersecting lines in our case is 34 degrees. 34 degrees is equal to the measure
of the intercepted arc πΆπΈ, which is 151 degrees, minus the measure of the
intercepted arc π΅π· and then divided it by two.

We need to solve for the measure of
the arc π΅π·. So, we multiply both sides of the
equation by two, which gives us 68 equals 151 minus the measure of arc π΅π·. So, we subtract 151 from both
sides, and we get negative 83 degrees equals the negative measure of arc π΅π·. We multiply both sides of the
equation by negative one and flip the sides, and we get the measure of arc π΅π·
equals positive 83 degrees. But this is not our final
answer. Weβre still trying to find the
measure of arc πΆπ΅.

But remember, we know that all of
these arcs must sum to 360 degrees. And we know that arc πΆπ΅ and arc
πΈπ· must be equal. We could say that theyβre equal to
π₯ degrees. If we do that, we could then create
the equation 151 plus π₯ plus π₯ plus 83 equals 360. If we combine like terms, 151 plus
83 equals 234. π₯ plus π₯ equals two π₯. Therefore, 234 plus two π₯ equals
360 degrees. We subtract 234 from both sides of
the equation, which tells us two π₯ equals 126. Divide both sides by two, and we
see that π₯ equals 63. This tells us that both arc πΆπ΅
and arc πΈπ· is equal to a measure of 63 degrees. We were primarily interested in the
measure of arc πΆπ΅, and we found that that measure is 63 degrees.