In the given figure, if m(arc AB) = 63 degrees, and m(arc BC) = 150 degrees, find m(arc AC).

Solution:

The sum of the arcs in a circle is 360 degrees. We are given the measures of two arcs, arc AB and arc BC. We need to find the measure of arc AC.

The entire circle can be represented by the sum of arcs AB, BC, and AC.

Therefore, m(arc AB) + m(arc BC) + m(arc AC) = 360 degrees.

Substitute the given values:

\( 63^{\circ} + 150^{\circ} + m(\text{arc AC}) = 360^{\circ} \)

Combine the known values:

\( 213^{\circ} + m(\text{arc AC}) = 360^{\circ} \)

To find m(arc AC), subtract 213 degrees from 360 degrees:

\( m(\text{arc AC}) = 360^{\circ} - 213^{\circ} \)

\( m(\text{arc AC}) = 147^{\circ} \)

Answer: The measure of arc AC is 147 degrees.