### Video Transcript

In the given figure, which of the
following inequalities is correct? (A) The measure of angle π΄π·π΅ is
less than the measure of angle π΄πΆπ΅. (B) The measure of angle π΄π΅π· is
greater than the measure of angle π΅π·πΆ. Option (C) the measure of angle
πΆπ΅π· is greater than the measure of angle πΆπ΅π΄. (D) The measure of angle π΄π·π΅ is
greater than the measure of angle π΄πΆπ΅. Or lastly, option (E) the measure
of angle π΅π΄πΆ is greater than the measure of angle π΅π·πΆ.

Weβre given a triangle π΄π΅πΆ,
where the angle at π΅ is split by a projection onto the side π΄πΆ at π·. And weβre asked to compare the
measures of various angles in the resulting triangles. We donβt actually have any of the
angle measures. But we can solve this by
considering the relationships between the various angles.

Letβs approach this by going
through each of the given options one by one, starting with option (A). This states that the measure of
angle π΄π·π΅ is less than the measure of angle π΄πΆπ΅. We see that angle π΄π·π΅ is an
exterior angle at π· to triangle πΆπ·π΅. And we know that the measure of any
exterior angle of a triangle is greater than the measure of either of the two
nonadjacent interior angles in that triangle. In the green triangle shown, this
means that angle π has greater measure than either angle π or angle π.

Applied to our triangle πΆπ·π΅ and
the exterior angle π΄π·π΅, this means that the measure of angle π΄π·π΅ must be
greater than the measure of angle π·πΆπ΅. Now angle π·πΆπ΅ is identical to
angle π΄πΆπ΅. So we see that, in fact, the
measure of angle π΄π·π΅ is greater than the measure of angle π΄πΆπ΅. This contradicts statement (A),
which says that the measure of angle π΄π·π΅ is less than the measure of angle
π΄πΆπ΅. So we can eliminate option (A),
which is incorrect.

Now considering option (B), this
claims that the measure of angle π΄π΅π· is greater than the measure of angle
π΅π·πΆ. From the figure, it looks as though
itβs the other way around. And we can show that this is the
case by again using the exterior angle property. Angle π΅π·πΆ is an exterior angle
at π· to the triangle π΄π·π΅. The two nonadjacent interior angles
to this exterior angle are angles π·π΄π΅ and π΄π΅π·. And by the exterior angle property,
the measure of angle π΅π·πΆ is greater than the measures of both angles π·π΄π΅ and
π΄π΅π·. This contradicts the claim in
option (B) that the measure of angle π΄π΅π· is greater than the measure of angle
π΅π·πΆ. Hence, weβve shown that the
statement in option (B) is incorrect. And we can eliminate option
(B).

Now moving on to option (C), this
says that angle πΆπ΅π· has measure greater than the measure of angle πΆπ΅π΄. We can see immediately that this
inequality cannot be true. This is because the measure of
angle πΆπ΅π΄ is the sum of the measures of the two angles πΆπ΅π· and π΄π΅π·. Since neither of these angles are
the zero angle, the measure of angle πΆπ΅π΄ must be greater than either one of
them. In particular, the measure of angle
πΆπ΅π΄ is greater than the measure of angle πΆπ΅π·, which contradicts option
(C). Hence, we can eliminate option
(C).

Now moving on to option (D), this
states that the measure of angle π΄π·π΅ is greater than the measure of angle
π΄πΆπ΅. We noted previously that angle
π΄π·π΅ is an exterior angle to the triangle π·πΆπ΅ at π· and hence by the exterior
angle property that angles πΆπ΅π· and π·πΆπ΅ must have measures less than that of
this exterior angle π΄π·π΅. But angles π·πΆπ΅ and π΄πΆπ΅ are
identical so that the measure of angle π΄πΆπ΅ must also be less than the measure of
angle π΄π·π΅, which is exactly the statement in option (D). Hence, option (D) is correct. The measure of angle π΄π·π΅ is
greater than the measure of angle π΄πΆπ΅.

Letβs look finally at option (E)
and see if we can eliminate this. Option (E) states that the measure
of angle π΅π΄πΆ is greater than the measure of angle π΅π·πΆ. To determine whether this is the
case or not, we can refer again to our exterior angle property. Noting once more that angle π΅π·πΆ
is an exterior angle to triangle π·π΄π΅, by our exterior angle property, we have
that the measure of exterior angle π΅π·πΆ is greater than the measure of interior
nonadjacent angle π·π΄π΅. And since angles π΅π΄πΆ and π·π΄π΅
are identical, we see that in fact the measure of angle π΅π·πΆ is greater than the
measure of angle π΅π΄πΆ, contradicting the statement in option (E). Hence, we can discount option
(E).

This means that only option (D) is
correct. The measure of angle π΄π·π΅ is
greater than the measure of angle π΄πΆπ΅.