Video: Finding the Relation between the Sides and Their Corresponding Angles in a Triangle Using Another Relation

From the figure below, determine the correct inequality from the following. [A] 𝐴𝐵 >𝐶𝐵 [B] 𝐴𝐵< 𝐶𝐵 [C] 𝐴𝐵 >𝐴𝐶 [D] 𝐴𝐶< 𝐶𝐵.

04:13

Video Transcript

From the figure below, determine the correct inequality from the following: 𝐴𝐵 is greater than 𝐶𝐵, 𝐴𝐵 is less than 𝐶𝐵, 𝐴𝐵 is greater than 𝐴𝐶, or 𝐴𝐶 is less than 𝐶𝐵.

Looking at the diagram, we can see that 𝐴𝐵, 𝐶𝐵, and 𝐴𝐶 all represent the lengths of sides of a triangle. We’ve been given four possibilities for relationships that could exist between the lengths of different pairs of sides. We haven’t been given any lengths in the diagram. Instead, we’ve been given some information about some of the angles. This suggests that we need to consider the relationship between the lengths of sides and the size of angles in a triangle. And therefore, we’re going to approach this question using the angle–side triangle inequality.

Here’s what the angle–side triangle inequality tells us. If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. Basically, what this means is that the longest side of a triangle is opposite the largest angle. The shortest side is opposite the smallest angle. And the middle side is opposite the middle angle. In the diagram, however, we’ve only currently got the size of one of the angles in the triangle. So we need to consider how we can find the other angles.

First of all, let’s consider angle 𝐴𝐵𝐶. We can see that the lines 𝐴𝐷 and 𝐶𝐵 are parallel as they’ve been marked with blue arrows on their lengths. The line 𝐴𝐵 is a transversal through these parallel lines. And therefore, we can see that the angle 𝐴𝐵𝐶 and the angle of 66 degrees are alternate interior angles. Which means that they’re congruent. So angle 𝐴𝐵𝐶 is also 66 degrees.

Now that we know the measures of two of the angles in the triangle, we can calculate the third because the angle sum in a triangle is always 180 degrees. So angle 𝐴𝐶𝐵 can be found by subtracting 52 degrees and 66 degrees from 180 degrees. It’s 62 degrees.

So now that we know the sizes of all three angles in the triangle, we can deduce something about the lengths of the three sides. The largest angle in the triangle is 66 degrees. And the angle–side triangle inequality tells us that the longest side of the triangle will be opposite this angle. So the longest side of the triangle is the side 𝐴𝐶. The second biggest angle in the triangle is the angle of 62 degrees which is opposite the side 𝐴𝐵. This means then that 𝐴𝐵 is the second longest side of the triangle. The smallest angle of 52 degrees is opposite the shortest side of the triangle. So 𝐶𝐵 is the shortest side.

Now that we have the three sides of the triangle ordered from longest to shortest, we can turn our attention to the four inequalities and determining which are true. Firstly, is 𝐴𝐵 greater than 𝐶𝐵? Yes, 𝐴𝐵 appears above 𝐶𝐵 in the list. Which means this first inequality is true. Is 𝐴𝐵 less than 𝐶𝐵? Well, this is the reverse of the inequality that we’ve just shown to be true. Therefore, this one must be false. Thirdly, is 𝐴𝐵 greater than 𝐴𝐶? No, 𝐴𝐶 is the longest side of the triangle. So this inequality is also false. And finally, is 𝐴𝐶 less than 𝐶𝐵? Again, this is false. 𝐴𝐶 is the longest side of the triangle.

So we can conclude that of the four inequalities, only one is true. 𝐴𝐵 is greater than 𝐶𝐵.

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