Video Transcript
Find the measure of angle
𝐴𝐶𝐷.
And we have a circle with points
𝐴, 𝐵, 𝐶, 𝐷 that lie on the circumference of that circle. We’re also given the measure of arc
𝐵𝐶 as being equal to 72 degrees. So, we’re looking to calculate the
size of angle 𝐴𝐶𝐷. That’s this one. And we’ll call that 𝜃 degrees. We need to find a way to link an
angle that lies on the circumference of the circle with an angle measured from the
center. And so we recall one of the key
circle theorems that looks at inscribed angles. It says that an inscribed angle is
half of the central angle that subtends the same arc. And this circle theorem is often
accompanied by this arrowhead-type shape. We notice the angle at the center,
two 𝑥, is double the angle at the circumference, or the angle at the circumference
is double the angle at the center.
Now, the key here is that these
angles must be subtended by the same arc. So here we’re subtending arc
𝐴𝐵. Now, our missing angle 𝐴𝐶𝐷 is
subtended by arc 𝐴𝐷. This means it’s going to be half of
the central angle subtended by the same arc. And since angle 𝐴𝐶𝐷 is 𝜃, we
can call that two 𝜃. So, how do we find the value of 𝜃
given this information? Well, we’re going to use the fact
that vertically opposite angles are equal to one another. In other words, when we have two
intersecting lines as below, 𝑦 is equal to 𝑦. In this case then, we can say that
two 𝜃 degrees must be equal to 72 degrees. These angles are vertically
opposite to one another. We can then solve for 𝜃 by
dividing through by two. And that means that 𝜃 degrees must
be equal to 72 divided by two, which is 36 degrees.
And remember, we defined the
measure of angle 𝐴𝐶𝐷 to be equal to 𝜃 degrees. And so the measure of angle 𝐴𝐶𝐷
is 36 degrees.
