Complete the following using less than, equal to, or greater than. If, in the triangle 𝐷𝐸𝐹, 𝐷𝐸 is greater than 𝐸𝐹, then the measure of angle 𝐹 blank the measure of angle 𝐷.
To help us solve this problem, I’ve actually sketched a triangle just to give us an understanding of our sides and our angles and which side will be opposite which angle.
First of all, we can see that 𝐷𝐸 is opposite angle 𝐹. And also side 𝐸𝐹 is opposite angle 𝐷. And these are the two sides that we’re interested in because we have this inequality here that tells us that side 𝐷𝐸 is greater than side 𝐸𝐹. But how is this gonna be useful for finding out whether the measure of angle 𝐹 is greater than, less than, or equal to the measure of angle 𝐷?
Well, actually, what we have is the angle–side relationship or angle–side relationship theorem. And this is actually something that can actually help us see which angle and which side are gonna be greater in size than the other.
And the angle–side relationship tells us that, in a triangle, the side opposite the larger angle is the longer side. So therefore, well, we know that the longer side is 𝐷𝐸 because 𝐷𝐸 is greater than 𝐸𝐹. So therefore, the angle that is opposite 𝐷𝐸 must be greater than the angle that is opposite 𝐸𝐹.
So from this, we can say that the measure of angle 𝐹 is gonna be greater than the measure of angle 𝐷. And this is the case because side 𝐷𝐸 is opposite angle 𝐹 and side 𝐷𝐸 is greater than side 𝐸𝐹 and side 𝐸𝐹 is opposite angle 𝐷.