Given that the line segment 𝐴𝐵 is
a diameter of the circle and line segment 𝐷𝐶 is parallel to line segment 𝐴𝐵,
find the measure of angle 𝐴𝐸𝐷.
We’re interested in the measure of
angle 𝐴𝐸𝐷; that’s this measure. And we’ve been given a few other
pieces of information. We know line segment 𝐷𝐶 is
parallel to line segment 𝐴𝐵. We know line segment 𝐴𝐵 is the
diameter. And on the figure, angle 𝐶𝐵𝐴 has
been labeled as 68.5 degrees.
At first, it might not seem like
there’s a very clear direction for where to go here. But if we start with the measure of
angle 𝐶𝐴𝐵, using that information, we could find the measure of arc 𝐶𝐴. Since angle 𝐶𝐴𝐵 is an inscribed
angle, its arc will be two times the measure of that inscribed angle. Arc 𝐴𝐶 will then be equal to two
times 68.5, which is 137 degrees. And because we know that line
segment 𝐴𝐵 is a diameter, arc 𝐴𝐵 must be equal to 180 degrees. We can also say that arc 𝐴𝐵 will
be equal to arc 𝐵𝐶 plus arc 𝐶𝐴.
We know 𝐴𝐵 needs to be 180
degrees and arc 𝐶𝐴 is 137 degrees. To solve for the measure of arc
𝐵𝐶, we can subtract 137 from both sides of the equation. And we get the measure of arc 𝐵𝐶
is 43 degrees. And here’s where our parallel
chords come into play. When you have parallel chords,
their intercepted arcs are going to be congruent. And that means because arc 𝐶𝐵
equals 43 degrees, arc 𝐷𝐴 also equals 43 degrees.
And at this point, we began to see
that arc 𝐷𝐴 is subtended by the angle 𝐴𝐸𝐷. Since angle 𝐴𝐸𝐷 is an inscribed
angle, its angle measure, the measure of angle 𝐴𝐸𝐷, is going to be equal to
one-half the measure of arc 𝐴𝐷. We know that the measure of arc
𝐴𝐷 is 43 degrees, and one-half of 43 is 21.5. And so, we can say that the measure
of angle 𝐴𝐸𝐷 is 21.5 degrees.