### Video Transcript

In the following figure, π΄π΅πΆ is
an equilateral triangle inscribed in a circle. Segment π΄π΅ and segment π·πΈ are
two chords of the circle, and line πΉπΊ is a tangent to the circle at πΉ. If segment π·πΈ is parallel to
segment π΄π΅, which is parallel to line πΉπΊ, and the measure of arc πΆπ· is 79
degrees, find the measure of πΈπ΅. Find the measure of π·πΉ.

The first thing we can do is add in
the information we know into our circle. π΄π΅πΆ is an equilateral
triangle. Therefore, each of their interior
angles measures 60 degrees. We also need to note that π·πΈ is
parallel to π΄π΅, which is parallel to πΉπΊ. Because we know that these lines
are parallel, we can also say the arcs between them will be equal in measure. This means the measure of arc π·π΄
will be equal to the measure of arc πΈπ΅. And the measure of arc π΄πΉ will be
equal to the measure of arc πΉπ΅. We know the first two chords will
be equal in measure by the measure of arcs between parallel chords theorem and the
second set, arc π΄πΉ and arc πΉπ΅, by the measure of arcs between a parallel chord
and tangent theorem. We know that arc πΆπ· equals 79
degrees.

The first arc we need to find the
measure of is πΈπ΅. To do that, letβs consider the
inscribed angle πΆπ΅π΄. Using this inscribed angle, we can
identify the measure of arc π΄πΆ. The arc π΄πΆ is two times the
measure of this inscribed angle, which we know is 60 degrees. Therefore, the arc π΄πΆ measures
120 degrees. We know that the larger arc π΄πΆ is
made up of the two smaller arcs π΄π· and π·πΆ. Therefore, we can say that 120
degrees equals the measure of arc π΄π· plus 79 degrees. Subtracting 79 degrees from both
sides, we find the measure of arc π΄π· to be 41 degrees. And weβve already shown that the
measure of arc πΈπ΅ will be equal to the measure of arc π΄π· by the measure of arcs
between parallel chords theorem. This makes the measure of arc πΈπ΅
equal to 41 degrees.

Our next unknown arc measure that
weβre looking for is π·πΉ. To do that, letβs first look at the
inscribed angle π΄πΆπ΅. The inscribed angle π΄πΆπ΅ subtends
the arc π΄π΅. Therefore, we can say the measure
of arc π΄π΅ equals two times the measure of angle π΄πΆπ΅, which again will be 120
degrees. Since the measure of arc π΄π΅
equals the measure of arc π΄πΉ plus the measure of arc πΉπ΅ and since arc πΉπ΅ and
arc π΄πΉ will be equal in measure, we can substitute two times the measure of arc
π΄πΉ in this equation. 120 degrees equals two times the
measure of arc π΄πΉ. The measure of arc π΄πΉ equals 60
degrees, which of course means that arc πΉπ΅ also measures 60 degrees.

Back to our unknown arc, thatβs
π·πΉ, the measure of arc π·πΉ equals the measure of arc π·π΄ plus the measure of arc
π΄πΉ, which is 41 degrees plus 60 degrees, 101 degrees.