### Video Transcript

If πΉ is a point on the line
between π΄ and π΅ and πΈ is a point on the line between πΆ and π·, the line between
π΄ and π΅ is perpendicular to the line between πΈ and πΉ, and the line between πΆ
and π· is perpendicular to the line between πΈ and πΉ, which of the following is
correct? Option (A) the line between π΄ and
π΅ is parallel to the line between πΆ and π·. Option (B) line segment πΈπΉ
bisects line segment π΄π΅. Option (C) the line between π΄ and
π΅ is parallel to the line between πΈ and πΉ. Option (D) the line between π΄ and
π΅ is perpendicular to the line between πΆ and π·. Or is it option (E) line segment
πΈπΉ bisects line segment πΆπ·?

In this question, we are given some
information about some lines and asked to determine which of five given options must
be true. To do this, letβs start by
sketching the given information, that is, that the line between π΄ and π΅ is
perpendicular to the line between πΈ and πΉ and that the line between πΆ and π· is
also perpendicular to this line, where πΈ and πΉ lie on the lines as shown. In our sketch, we see that the line
between πΈ and πΉ is a transversal of the other two lines. And we see that the corresponding
angles of this transversal both have equal measure since they are both right
angles.

This is enough to confirm that the
line between π΄ and π΅ and the line between πΆ and π· must be parallel since the
corresponding angles of the transversal have equal measure. Alternatively, we can use the
specific result that any two lines perpendicular to the same line are parallel. Hence, the answer is option (A):
the line between π΄ and π΅ is parallel to the line between πΆ and π·.

It is worth noting that we can show
that none of the other options must be true using our sketch. First, we can choose points πΈ and
πΉ anywhere on the lines. So, we can choose point πΉ to not
be the midpoint of line segment π΄π΅. Therefore, option (B) is not true
in general. The same reasoning works to show
that option (E) is not true in general, since we can choose point πΈ to not be the
midpoint of the line segment πΆπ·. We are told that the line segment
between π΄ and π΅ is perpendicular to the line between πΈ and πΉ, so they are not
parallel. Therefore, option (C) is not
true.

Finally, since we have shown that
the line between π΄ and π΅ is parallel to the line between πΆ and π·, we can
conclude that they do not intersect and so they are not perpendicular. Hence, the only statement that is
true in general is option (A).