### Video Transcript

Consider triangle π΄π΅πΆ and
lines π΄π and πΈπ·, which are parallel to line πΆπ΅. Find the length of the line
segment π΄π΅. Find the measure of angle
π΄π΅πΆ.

We are given three parallel
lines and two transversals of these lines. We can then recall that if a
set of parallel lines divide a transversal into segments of equal length, then
they divide any other transversal into segments of equal length. Since π΄πΈ is equal to πΈπΆ,
the segments of the other transversal must be equal in length. So π΄π· is equal to π·π΅, which
equals five millimeters. Since π΄π΅ is equal to π΄π·
plus π·π΅, then π΄π΅ equals five millimeters plus five millimeters, which equals
10 millimeters. π΄π΅ is equal to 10
millimeters.

It appears in the diagram that
triangle π΄π΅πΆ is a right triangle. However, we need to justify why
this is the case. We can do this by recalling
that if a line is perpendicular to line πΏ, then it is perpendicular to any line
parallel to πΏ. Since line πΈπ· is
perpendicular to line π΄πΆ and line πΈπ· is parallel to line π΅πΆ, we must have
that lines π΅πΆ and π΄πΆ are perpendicular. This means the angle at πΆ has
a measure of 90 degrees, so π΄π΅πΆ is a right triangle.

The sum of the measures of the
interior angles in a triangle is 180 degrees. So 180 degrees equals 35
degrees plus 90 degrees plus the measure of angle π΄π΅πΆ. Rearranging the equation, we
have the measure of angle π΄π΅πΆ equals 180 degrees minus 35 degrees minus 90
degrees, which equals 55 degrees. The answers to the two parts of
this question are 10 millimeters and 55 degrees.