# Question Video: Comparing Interior and Exterior Angles of Triangles Mathematics

In the figure, which of the following inequalities is correct? [A] πβ π΄π·π΅ < πβ π΄πΆπ΅ [B] πβ π΄π΅π· > πβ π΅π·πΆ [C] πβ πΆπ΅π· > πβ πΆπ΅π΄ [D] πβ π΄π·π΅ > πβ π΄πΆπ΅ [E] πβ π΅π΄πΆ > πβ π΅π·πΆ

05:18

### 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΄πΆπ΅. (B) The measure of angle π΄π΅π· is greater than the measure of angle π΅π·πΆ. Option (C) the measure of angle πΆπ΅π· is greater than the measure of angle πΆπ΅π΄. (D) The measure of angle π΄π·π΅ is greater than the measure of angle π΄πΆπ΅. Or lastly, option (E) the measure of angle π΅π΄πΆ is greater than the measure of angle π΅π·πΆ.

Weβre given a triangle π΄π΅πΆ, where the angle at π΅ is split by a projection onto the side π΄πΆ at π·. And weβre asked to compare the measures of various angles in the resulting triangles. We donβt actually have any of the angle measures. But we can solve this by considering the relationships between the various angles.

Letβ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 π΄π·π΅ is less than the measure of angle π΄πΆπ΅. We see that angle π΄π·π΅ is an exterior angle at π· to triangle πΆπ·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 πΆπ·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·πΆπ΅. Now angle π·πΆπ΅ is identical to angle π΄πΆπ΅. So we see that, in fact, the measure of angle π΄π·π΅ is greater than the measure of angle π΄πΆπ΅. This contradicts statement (A), which says that the measure of angle π΄π·π΅ is less than the measure of angle π΄πΆπ΅. 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·πΆ. From the figure, it looks as though itβs the other way around. And we can show that this is the case by again using the exterior angle property. Angle π΅π·πΆ 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·πΆ 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·πΆ. Hence, weβ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 πΆπ΅π· has measure greater than the measure of angle πΆπ΅π΄. We can see immediately that this inequality cannot be true. This is because the measure of angle πΆπ΅π΄ is the sum of the measures of the two angles πΆπ΅π· and π΄π΅π·. Since neither of these angles are the zero angle, the measure of angle πΆπ΅π΄ must be greater than either one of them. In particular, the measure of angle πΆπ΅π΄ is greater than the measure of angle πΆπ΅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΄πΆπ΅. We noted previously that angle π΄π·π΅ is an exterior angle to the triangle π·πΆπ΅ at π· and hence by the exterior angle property that angles πΆπ΅π· and π·πΆπ΅ must have measures less than that of this exterior angle π΄π·π΅. But angles π·πΆπ΅ and π΄πΆπ΅ are identical so that the measure of angle π΄πΆπ΅ 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΄πΆπ΅.

Letβs look finally at option (E) and see if we can eliminate this. Option (E) states that the measure of angle π΅π΄πΆ is greater than the measure of angle π΅π·πΆ. 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·πΆ is an exterior angle to triangle π·π΄π΅, by our exterior angle property, we have that the measure of exterior angle π΅π·πΆ is greater than the measure of interior nonadjacent angle π·π΄π΅. And since angles π΅π΄πΆ and π·π΄π΅ are identical, we see that in fact the measure of angle π΅π·πΆ is greater than the measure of angle π΅π΄πΆ, 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΄πΆπ΅.