Which is the correct relationship between 𝐴𝐶 and 𝐴𝐵, a) where 𝐴𝐶 is less than 𝐴𝐵, b) where 𝐴𝐶 is greater than 𝐴𝐵, or c) where 𝐴𝐶 is equal to 𝐴𝐵?
To help us solve this problem, we’re actually gonna use something called the angle-side relationship theorem. And the angle-side relationship theorem says that in a triangle, the side opposite the larger angle is the longer side. So therefore, the angles that we want to find are 𝐴𝐵𝐶, because this is opposite 𝐴𝐶, and the angle 𝐴𝐶𝐵, because this is opposite side 𝐴𝐵.
Okay, so let’s go on and find some angles. So first of all, we’re gonna start with finding the angle 𝐴𝐶𝐵. Well angle 𝐴𝐶𝐵 is gonna be equal to 61 degrees. However, we do need to give reasoning for this. Well, if we have a look at the diagram, we can see that we have two parallel sides. And these are shown by the arrows.
So therefore, as they are parallel lines, we can say that angle 𝐴𝐶𝐵 is equal to 61 degrees because alternate angles are equal. And I’ve actually highlighted the alternate angle with a little red mark so we can see that 61 degrees and the 𝐴𝐶𝐵 are going to be the same. Sometimes they’re called 𝑍 angles because of the shape they make.
Okay, so now we found angle 𝐴𝐶𝐵. Let’s move on and find angle 𝐴𝐵𝐶. Well again, when we’re trying to find angle 𝐴𝐵𝐶, what we want to do is we actually want to look at the fact that it’s a pair of parallel lines. And due to this, we can say that angle 𝐴𝐵𝐶 is equal to 67 degrees. And we can say this because angle 𝐴𝐵𝐶 and angle 𝐷𝐴𝐸 are corresponding angles.
And therefore, because they’re corresponding, they’re going to be equal. And I’ve just shown a little sketch, what angles would be considered corresponding. And as you can see, that’s like our angle 𝐴𝐵𝐶 and angle 𝐷𝐴𝐸. Okay, great! So we’ve now found out the two angles, 61 degrees and 67 degrees. So then we just need to remind ourselves that angle 𝐴𝐶𝐵 is opposite 𝐴𝐵 and angle 𝐴𝐵𝐶 is opposite 𝐴𝐶.
And we know that angle 𝐴𝐵𝐶 is greater than 𝐴𝐶𝐵. And we know this because 67 degrees is greater than 61 degrees. So therefore, if we look at the angle-side relationship theorem, where it says that in a triangle the side opposite the larger angle is the longer side, we can say that 𝐴𝐶 is greater than 𝐴𝐵. So therefore, b is the correct relationship between 𝐴𝐶 and 𝐴𝐵 because, as we said, it’s where 𝐴𝐶 is greater than 𝐴𝐵. And this is because angle 𝐴𝐵𝐶, which is opposite 𝐴𝐶, is greater than angle 𝐴𝐶𝐵, which is opposite 𝐴𝐵.