Video Transcript
If the measure of angle 𝐶𝐵𝐷
equals 20 degrees, find the measure of angle 𝐵𝐴𝐶.
We begin by recalling the
definition of an inscribed angle. An angle is inscribed in a circle
if it is formed by the intersection of two chords on the circumference. According to this definition,
𝐵𝐴𝐶 and 𝐶𝐵𝐷 are both inscribed angles. From the diagram, we see that
inscribed angle 𝐵𝐴𝐶, highlighted in blue, is subtended by arc 𝐵𝐶, which is
highlighted in pink. And inscribed angle 𝐶𝐵𝐷, which
is highlighted in orange, is subtended by arc 𝐶𝐷, which is highlighted in
green.
Next, we notice that the diagram
shows arc 𝐵𝐶 is congruent to arc 𝐶𝐷. We recall a property of inscribed
angles, which says that all inscribed angles subtended by congruent arcs in a circle
are equal in measure. We have arc 𝐵𝐶 is congruent to
arc 𝐶𝐷. So, according to this property,
their inscribed angles must have equal measure. That means the measure of inscribed
angle 𝐵𝐴𝐶 equals the measure of inscribed angle 𝐶𝐵𝐷. We know that the measure of angle
𝐶𝐵𝐷 is 20 degrees. Hence, by substitution, we find
that the measure of angle 𝐵𝐴𝐶 equals 20 degrees as well.
We may also recall that the measure
of an inscribed angle equals half the measure of the arc it is subtended by. In other words, an intercepted arc
has twice the measure of its inscribed angle. We are given the measure of angle
𝐶𝐵𝐷 equals 20 degrees. So, in this case, the measure of
the intercepted arc 𝐶𝐷 must be 40 degrees. And because arc 𝐵𝐶 is congruent
to arc 𝐶𝐷, arc 𝐵𝐶 must also measure 40 degrees.
Therefore, because angle 𝐵𝐴𝐶 is
subtended by arc 𝐵𝐶, we take half of 40 degrees to find the inscribed angle
measure. This supports our final answer of
20 degrees.
