Question Video: Comparing Interior and Exterior Angles of Triangles Mathematics

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.