Video Transcript
Find the measure of angle
𝐴𝐵𝐶.
Let’s begin by identifying the
angle whose measure we’re asked to find. It’s the angle formed when we move
from 𝐴 to 𝐵 to 𝐶. So that’s this angle here on the
figure. Now this is an inscribed angle on
the circle’s circumference. So we know that its measure will be
one-half of its intercepted arc. Its intercepted arc is the arc
𝐴𝐶. So we have the equation the measure
of the angle 𝐴𝐵𝐶 is equal to one-half the measure of the arc 𝐴𝐶.
Let’s consider then how we might be
able to calculate the measure of this arc. We can see that the other
information given in the question is, firstly, the angle formed by the intersection
of two chords inside a circle, the chords 𝐴𝐵 and 𝐶𝐷. We’re also told the measure of the
arc intercepted by this angle, the measure of the arc 𝐵𝐷, which is 98 degrees. The angles of intersecting chords
theorem tells us that the measure of the angle between two chords that intersect
inside a circle is half the sum of the measures of the arcs intercepted by the angle
and its vertical angle. The arc intercepted by the angle of
88 degrees is the arc 𝐵𝐷, and the arc intercepted by its vertical angle is the arc
𝐴𝐶. So we can form an equation 88
degrees is equal to one-half the measure of the arc 𝐵𝐷 plus the measure of the arc
𝐴𝐶.
Remember though that we know the
measure of the arc 𝐵𝐷. It’s given to us in the figure as
98 degrees. So we can substitute this value
into our equation, and we’ll then be able to solve to find the measure of the arc
𝐴𝐶. We have 88 degrees is equal to
one-half of 98 degrees plus the measure of the arc 𝐴𝐶. Multiplying each side of the
equation by two, we have 176 degrees equals 98 degrees plus the measure of the arc
𝐴𝐶. And finally, subtracting 98 degrees
from each side, we find that the measure of the arc 𝐴𝐶 is 78 degrees.
The final step in this problem is
to take this value for the measure of the arc 𝐴𝐶 and substitute it into our first
equation. We have then that the measure of
the angle 𝐴𝐵𝐶, which is half the measure of its intercepted arc, is one-half
multiplied by 78 degrees, which is 39 degrees. So, by recalling the relationship
between the measures of an inscribed angle and its intercepted arc and also the
angles of intersecting chords theorem, we’ve found that the measure of the angle
𝐴𝐵𝐶 is 39 degrees.
