Video Transcript
In the given figure, which of the
following inequalities is correct? (A) The measure of angle 𝐴𝐷𝐵 is
less than the measure of angle 𝐴𝐶𝐵. (B) The measure of angle 𝐴𝐵𝐷 is
greater than the measure of angle 𝐵𝐷𝐶. Option (C) the measure of angle
𝐶𝐵𝐷 is greater than the measure of angle 𝐶𝐵𝐴. (D) The measure of angle 𝐴𝐷𝐵 is
greater than the measure of angle 𝐴𝐶𝐵. Or lastly, option (E) the measure
of angle 𝐵𝐴𝐶 is greater than the measure of angle 𝐵𝐷𝐶.
We’re given a triangle 𝐴𝐵𝐶,
where the angle at 𝐵 is split by a projection onto the side 𝐴𝐶 at 𝐷. And we’re asked to compare the
measures of various angles in the resulting triangles. We don’t actually have any of the
angle measures. But we can solve this by
considering the relationships between the various angles.
Let’s approach this by going
through each of the given options one by one, starting with option (A). This states that the measure of
angle 𝐴𝐷𝐵 is less than the measure of angle 𝐴𝐶𝐵. We see that angle 𝐴𝐷𝐵 is an
exterior angle at 𝐷 to triangle 𝐶𝐷𝐵. 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 𝑑 has greater measure than either angle 𝑎 or angle 𝑏.
Applied to our triangle 𝐶𝐷𝐵 and
the exterior angle 𝐴𝐷𝐵, this means that the measure of angle 𝐴𝐷𝐵 must be
greater than the measure of angle 𝐷𝐶𝐵. Now angle 𝐷𝐶𝐵 is identical to
angle 𝐴𝐶𝐵. So we see that, in fact, the
measure of angle 𝐴𝐷𝐵 is greater than the measure of angle 𝐴𝐶𝐵. This contradicts statement (A),
which says that the measure of angle 𝐴𝐷𝐵 is less than the measure of angle
𝐴𝐶𝐵. So we can eliminate option (A),
which is incorrect.
Now considering option (B), this
claims that the measure of angle 𝐴𝐵𝐷 is greater than the measure of angle
𝐵𝐷𝐶. From the figure, it looks as though
it’s the other way around. And we can show that this is the
case by again using the exterior angle property. Angle 𝐵𝐷𝐶 is an exterior angle
at 𝐷 to the triangle 𝐴𝐷𝐵. The two nonadjacent interior angles
to this exterior angle are angles 𝐷𝐴𝐵 and 𝐴𝐵𝐷. And by the exterior angle property,
the measure of angle 𝐵𝐷𝐶 is greater than the measures of both angles 𝐷𝐴𝐵 and
𝐴𝐵𝐷. This contradicts the claim in
option (B) that the measure of angle 𝐴𝐵𝐷 is greater than the measure of angle
𝐵𝐷𝐶. Hence, we’ve 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 𝐶𝐵𝐷 has measure greater than the measure of angle 𝐶𝐵𝐴. We can see immediately that this
inequality cannot be true. This is because the measure of
angle 𝐶𝐵𝐴 is the sum of the measures of the two angles 𝐶𝐵𝐷 and 𝐴𝐵𝐷. Since neither of these angles are
the zero angle, the measure of angle 𝐶𝐵𝐴 must be greater than either one of
them. In particular, the measure of angle
𝐶𝐵𝐴 is greater than the measure of angle 𝐶𝐵𝐷, which contradicts option
(C). Hence, we can eliminate option
(C).
Now moving on to option (D), this
states that the measure of angle 𝐴𝐷𝐵 is greater than the measure of angle
𝐴𝐶𝐵. We noted previously that angle
𝐴𝐷𝐵 is an exterior angle to the triangle 𝐷𝐶𝐵 at 𝐷 and hence by the exterior
angle property that angles 𝐶𝐵𝐷 and 𝐷𝐶𝐵 must have measures less than that of
this exterior angle 𝐴𝐷𝐵. But angles 𝐷𝐶𝐵 and 𝐴𝐶𝐵 are
identical so that the measure of angle 𝐴𝐶𝐵 must also be less than the measure of
angle 𝐴𝐷𝐵, which is exactly the statement in option (D). Hence, option (D) is correct. The measure of angle 𝐴𝐷𝐵 is
greater than the measure of angle 𝐴𝐶𝐵.
Let’s look finally at option (E)
and see if we can eliminate this. Option (E) states that the measure
of angle 𝐵𝐴𝐶 is greater than the measure of angle 𝐵𝐷𝐶. To determine whether this is the
case or not, we can refer again to our exterior angle property. Noting once more that angle 𝐵𝐷𝐶
is an exterior angle to triangle 𝐷𝐴𝐵, by our exterior angle property, we have
that the measure of exterior angle 𝐵𝐷𝐶 is greater than the measure of interior
nonadjacent angle 𝐷𝐴𝐵. And since angles 𝐵𝐴𝐶 and 𝐷𝐴𝐵
are identical, we see that in fact the measure of angle 𝐵𝐷𝐶 is greater than the
measure of angle 𝐵𝐴𝐶, contradicting the statement in option (E). Hence, we can discount option
(E).
This means that only option (D) is
correct. The measure of angle 𝐴𝐷𝐵 is
greater than the measure of angle 𝐴𝐶𝐵.
