Video Transcript
Find the measure of angle
๐ด๐ต๐ถ.
Letโs begin by identifying the
angle whose measure weโre asked to find. Itโs the angle formed when we move
from ๐ด to ๐ต to ๐ถ. So thatโs this angle here on the
figure. Now this is an inscribed angle on
the circleโs circumference. So we know that its measure will be
one-half of its intercepted arc. Its intercepted arc is the arc
๐ด๐ถ. So we have the equation the measure
of the angle ๐ด๐ต๐ถ is equal to one-half the measure of the arc ๐ด๐ถ.
Letโs 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 ๐ด๐ต and ๐ถ๐ท. Weโre also told the measure of the
arc intercepted by this angle, the measure of the arc ๐ต๐ท, 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 ๐ต๐ท, and the arc intercepted by its vertical angle is the arc
๐ด๐ถ. So we can form an equation 88
degrees is equal to one-half the measure of the arc ๐ต๐ท plus the measure of the arc
๐ด๐ถ.
Remember though that we know the
measure of the arc ๐ต๐ท. Itโs given to us in the figure as
98 degrees. So we can substitute this value
into our equation, and weโll then be able to solve to find the measure of the arc
๐ด๐ถ. We have 88 degrees is equal to
one-half of 98 degrees plus the measure of the arc ๐ด๐ถ. Multiplying each side of the
equation by two, we have 176 degrees equals 98 degrees plus the measure of the arc
๐ด๐ถ. And finally, subtracting 98 degrees
from each side, we find that the measure of the arc ๐ด๐ถ is 78 degrees.
The final step in this problem is
to take this value for the measure of the arc ๐ด๐ถ and substitute it into our first
equation. We have then that the measure of
the angle ๐ด๐ต๐ถ, 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, weโve found that the measure of the angle
๐ด๐ต๐ถ is 39 degrees.
