Video Transcript
In the given figure, if the measure
of arc π΅π· equals 65 degrees, find the measure of arc πΆπ·.
In the diagram, we notice that
weβve been given a pair of parallel chords. That is, line segment π΄π΅ is
parallel to line segment πΆπ·. And we recall that arcs formed by a
pair of parallel chords are congruent. So arc π΄πΆ and arc π΅π· are
congruent, which means the measures of these arcs must be equal. And so we see that the measure of
arc π΄πΆ is 65 degrees.
Next, we see that line segment π΄π΅
in fact passes through the center of the circle. It must therefore be the diameter
of the circle. And so it splits this circle
exactly in half. And so we can say that the measure
of both arcs π΄π΅ are 180 degrees. Now, of course, weβre interested in
the portion of the circle which passes through points πΆ and π·. So weβve called that the measure of
arc π΄πΆπ·π΅. The question wants us to find the
measure of arc πΆπ·. And we now know that the measure of
all the individual arcs between π΄ and π΅ is 180 degrees. So we can say that the sum of the
measure of arc π΄πΆ, the measure of arc πΆπ·, and the measure of arc π·π΅ is
180. But remember, we said that arcs
π΄πΆ and π·π΅ are congruent and their measures are 65 degrees. So our equation becomes 65 degrees
plus the measure of arc πΆπ· plus another 65 degrees equals 180. And then we simplify that left-hand
side.
We can now solve this equation for
the measure of arc πΆπ· by subtracting 130 from both sides. So itβs 180 minus 130, which is of
course 50. So the measure of arc πΆπ· is 50
degrees.
