In the following figure, 𝑀 is a circle. Segment 𝐴𝐵 and segment 𝐶𝐷 are two chords of the circle, and line 𝐸𝐹 is a tangent to the circle at 𝐸. If segment 𝐴𝐵 is parallel to segment 𝐶𝐷 which is parallel to line 𝐸𝐹, the measure of arc 𝐴𝐶 equals 30 degrees, and the measure of arc 𝐷𝐸 equals 74 degrees, find the measure of arc 𝐴𝐵.
Let’s add some information we’re given to the figure. Arc 𝐴𝐶 measures 30 degrees. Arc 𝐷𝐸 measures 74 degrees. The key point here is that we’re working with parallel chords and a parallel tangent line. The measures of the arcs between parallel chords in a circle are equal. Therefore, arc 𝐵𝐷 will be equal in measure to arc 𝐴𝐶. And similarly, the measures of the arcs between a parallel chord and a tangent of that circle are equal. In our circle, that means that arc 𝐷𝐸 will be equal in measure to arc 𝐶𝐸. We can therefore add the measure of arc 𝐵𝐷 equals 30 degrees and the measure of arc 𝐶𝐸 equals 74 degrees onto our figure.
Our missing arc is 𝐴𝐵. We know that the sum of the measures of all the arcs in a circle must equal 360 degrees. Therefore, the measure of arc 𝐴𝐵 will be equal to 360 degrees minus the sum of all the other four arcs we know: 74 degrees plus 74 degrees plus 30 degrees plus 30 degrees. 360 degrees minus 208 degrees equals 152 degrees. The measure of arc 𝐴𝐵 in this figure is 152 degrees.