### Video Transcript

Find the measure of the minor arc πΆπ·, given that the measure of the minor arc π΄π· is equal to 168 degrees and the measure of the major arc π΅πΆ is 256 degrees.

The key to answering this question is to pay close attention to the distinction between a minor arc and a major arc between two points on any circle. For example, in the circle shown with two points π and π, there are two arcs that could be called arc ππ. The smaller of the two is called the minor arc ππ, and the larger of the two is called the major arc ππ. We should also note here that when weβre dealing with the same two points, the sum of the measures of the major and minor arcs of these two points is 360 degrees. In the given problem, we see that the minor arc π΄π· has a measure 168 degrees. And thatβs where the minor arc is the smaller of the two arcs between the two points π΄ and π·.

Similarly, weβre told that the measure of the major arc π΅πΆ is 256 degrees. And this corresponds to the larger of the two arcs between the points π΅ and πΆ on the circle as shown. Now our unknown is the minor arc πΆπ·. And before we do anything else, we should note that we have two parallel chords in our circle. And we know that the measure of arcs between parallel chords of a circle are equal. This means that the minor arcs πΆπ· and π΄π΅ in the given circle are equal in measure.

So how can we calculate the measure of these minor arcs? Well, one way of doing this is to calculate the minor arc π΅πΆ. We know that the major arc π΅πΆ has measure 256 degrees. Therefore, the minor arc π΅πΆ will have measure 360 degrees minus 256 degrees. And thatβs 104 degrees. And so the minor arc π΅πΆ has measure 104 degrees.

Now remember, the minor arc π΄π· has measure 168 degrees. So if we let the measure of the minor arcs π΄π΅ and πΆπ· be lowercase π, then we see that the minor arc π΄π· is made up of minor arc π΅πΆ, thatβs 104 degrees, plus two multiplied by lowercase π. And we know that the minor arc π΄π· has measured 168 degrees. So we can set up an equation as shown. We can solve this equation for π by subtracting 104 degrees from both sides, which gives us two π is 64 degrees. And if we then divide both sides by two, this gives us π is 32 degrees. And this means that both minor arcs πΆπ· and π΄π΅ have the measure 32 degrees. And this means that the measure of the minor arc πΆπ· is 32 degrees.

Now, if we wanted to go back and do an additional check, we could go back to the minor arc π΄π· and find the major arc π΄π·. That will be 360 degrees minus 168 degrees, which is 192 degrees. And once weβve labeled this on our diagram, we should find that 32 degrees, thatβs π, minor arc π΄π΅, plus 104 degrees, thatβs minor arc π΅πΆ, plus 32 degrees, thatβs π again, which is minor arc πΆπ·, plus 192 degrees, thatβs major arc π΄π·, should equal 360 degrees, which it does. And this confirms that the measure of the minor arc πΆπ· is 32 degrees.