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Triangle Side and Angle Inequalities - Problem 2
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The longest side of a triangle is opposite the largest interior angle, and the shortest side is opposite the smallest angle. As a result, without knowing the exact lengths of the sides of a triangle, it is possible to put them in order of length based solely on the given angle measurements. To put the sides in order of longest to shortest, first find the largest angle -- the longest side is opposite this angle. Then, find the second largest angle and the side opposite from it to find the next largest angle, and finally, the shortest side will be opposite the smallest angle.

If an angle from the triangle is missing, use the triangle angle sum theorem to solve for the measure of the missing angle.

We can use the size of an angle in a triangle to determine which side is the largest. In this problem it’s saying, list them in largest to shortest or longest to shortest and we’re given incomplete information. So before we can say well one side has to be larger than the other, we have to find our missing angle.

A lot of times in geometry you’re going to be given a problem where before you can really start there is something you need to do first. We just know that 67 plus 53 plus this missing angle is 180. So we can say that 67 plus 53 plus x, if I call that x is equal180 degrees. 67 and 53 is 120, so 120 plus x is equal 180 degrees and by subtracting 120 we see that our missing angle is 60 degrees. So I’m going to erase the x and I’m going to write 60 degrees.

So now we can go about solving the problem. The longest side, again you have to be careful, sometimes it’ll say shortest to longest, sometimes they’ll switch it around. The longest side will be opposite the largest angle so largest angle here is 67 degrees. 67 degrees is opposite the side NO, so NO is your longest side. Your next largest angle between 60 and 53 is 60 and 60 is opposite of the side MN. So MN is your next longest side, and finally 53 is opposite the side MO. So what do we do here, first we find our missing angle using the triangle sum and then we say the largest angle is opposite the longest side.

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