Find the measure of arc 𝐵𝐸, where 𝑀 is the center of the circle.
First, let’s identify arc 𝐵𝐸. After we highlight arc 𝐵𝐸, it’s important to notice that the segment 𝐵𝐴 and the segment 𝐸𝐶 are two parallel chords in this circle. We know by the measures of arcs between parallel chords theorem that the measure of the arcs between parallel chords of a circle are equal. This means that the arc 𝐵𝐸 will be equal in measure to the arc 𝐴𝐶. We can also see that line segment 𝐷𝐶 intersects line segment 𝐵𝐴 at the center of the circle 𝑀.
We can therefore say that angle 𝐴𝑀𝐶 is the opposite or vertical angle from angle 𝐵𝑀𝐷 and is, therefore, also equal to 39 degrees. Since 𝑀 is the center of the circle and arc 𝐴𝐶 is subtended by the angle 𝐴𝑀𝐶, we can say that the arc measure is equal to the central angle measure, which is 39 degrees. Since arc 𝐴𝐶 is 39 degrees, arc 𝐵𝐸 must also be equal to 39 degrees.