Video: Understanding the Triangle Inequality

Complete the following using <, =, or >: If, in the triangle ๐ท๐ธ๐น, ๐ท๐ธ > ๐ธ๐น, then ๐‘šโˆ ๐น ๏ผฟ ๐‘šโˆ ๐ท.

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Video Transcript

Complete the following using less than, equal to, or greater than. If, in the triangle ๐ท๐ธ๐น, ๐ท๐ธ is greater than ๐ธ๐น, then the measure of angle ๐น blank the measure of angle ๐ท.

To help us solve this problem, Iโ€™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 ๐ท๐ธ is opposite angle ๐น. And also side ๐ธ๐น is opposite angle ๐ท. And these are the two sides that weโ€™re interested in because we have this inequality here that tells us that side ๐ท๐ธ is greater than side ๐ธ๐น. But how is this gonna be useful for finding out whether the measure of angle ๐น is greater than, less than, or equal to the measure of angle ๐ท?

Well, actually, what we have is the angleโ€“side relationship or angleโ€“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 angleโ€“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 ๐ท๐ธ because ๐ท๐ธ is greater than ๐ธ๐น. So therefore, the angle that is opposite ๐ท๐ธ must be greater than the angle that is opposite ๐ธ๐น.

So from this, we can say that the measure of angle ๐น is gonna be greater than the measure of angle ๐ท. And this is the case because side ๐ท๐ธ is opposite angle ๐น and side ๐ท๐ธ is greater than side ๐ธ๐น and side ๐ธ๐น is opposite angle ๐ท.

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