### Video Transcript

Which inequality is satisfied by
this figure? (A) The measure of angle π΅ is less
than the measure of angle π΅π΄πΆ, which is less than the measure of angle πΆ. (B) The measure of angle π·π΄πΆ is
less than the measure of angle π΅, which is less than the measure of angle πΆ. (C) The measure of angle π΅π΄πΆ is
less than the measure of angle πΆ, which is less than the measure of angle
π·π΄πΆ. Or (D) the measure of angle π·π΄πΆ
is less than the measure of angle π΅, which is less than the measure of angle
π΅π΄πΆ. And finally (E) the measure of
angle πΆ is less than the measure of angle π΅, which is less than the measure of
angle π΅π΄πΆ.

For us to be able to order these
angles, weβll need to find the measures of a few of the missing angles. Currently, we donβt know the
measure of angle πΆ or the measure of angle π΅π΄πΆ. We should see that angle π΅π΄πΆ and
angle π·π΄πΆ make a straight line. If these two angles together make a
straight line, they are supplementary angles and theyβll add together to be 180
degrees. If we plug in the measure for angle
π·π΄πΆ, which we know is 92 degrees, then the measure of angle π΅π΄πΆ plus 92
degrees equals 180 degrees. And if we subtract 92 degrees from
both sides of this equation, then we find that the measure of angle π΅π΄πΆ equals 88
degrees. We can add that to our figure.

And from there, we recognize that
we have triangle π΄π΅πΆ. And in a triangle, the three angles
must add up to 180 degrees. So we say the measure of angle π΅
plus the measure of angle π΅π΄πΆ plus the measure of angle πΆ must equal 180
degrees. Angle π΅ is 52 degrees, angle
π΅π΄πΆ is 88 degrees, and we want to find angle πΆ. If we add 52 plus 88, we get 140
degrees. To find the measure of angle πΆ, we
then need to subtract 140 degrees from both sides of our equation to show that the
measure of angle πΆ is 40 degrees. And then we can add that back to
our figure.

What we can do now is list the
angles that we know in order from least to greatest. Our smallest angle is angle πΆ,
which measures 40 degrees, followed by the measure of angle π΅, which is 52 degrees,
followed by the measure of angle π΅π΄πΆ, which is 88 degrees. And the largest of the angles we
see in this figure is angle π·π΄πΆ, which is 92 degrees. Using this compound inequality, we
can see which of the answer choices is true. And only option (E) list the angles
in correct order, which says the measure of angle πΆ is less than the measure of
angle π΅, which is less than the measure of angle π΅π΄πΆ.