In the given diagram, we want to find m BC, which is a minor arc.
Note that the major arc BTC and the minor arc BC are adjacent arcs. By the Arc Addition Postulate, we know that the measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs. In this case, the arc formed by these two adjacent arcs is a full circle, which measures 360^(∘).
m BTC+m BC=360^(∘)To find the measure of BC, we have to find first m BTC. By looking at the diagram, we note that BTC is formed by the adjacent arcs BT and TC.
We can see in the diagram that the central angle for BT is a right angle. Recall that the measure of a minor arc is equal to the measure of its corresponding central angle.
mBT=m∠BPT ⇔ mBT=90^(∘)
In the same way, we see that the central angle for the minor arc TC is 128^(∘). By the Arc Addition Postulate, we know that the measure of the major arc BTC is the sum of the measures of BT and TC.
m BTC=mBT+mTC
Let's substitute 90^(∘) for mBT and 128^(∘) for mTC in the above equation, and solve for BTC.
