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Triangle Side and Angle Inequalities - Problem 3
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Given adjacent triangles and some known angles, it is possible to list the sides of those triangles in order of length. Recall that the shortest side of a triangle is opposite the smallest angle, and the longest side is opposite the largest angle.

When there are two triangles that are adjacent, first consider which of the two is larger. To determine this, remember various properties of triangles, such as the fact that the hypotenuse, or the side of a triangle opposite a right angle, is always the longest side. Using this, it is possible to find the relationship between sides of two adjacent triangles.

A complicated application of what we know about the relationship between sides and angles and their relative size within a given triangle is a problem like this, where you have more than one triangle and they share a common side. This problem is asking us to list the sides in order of smallest to largest.

So let’s start by saying which of these triangles is going to be larger? And as we can see this smaller triangle right here has W as its hypotenuse. The hypotenuse is opposite the 90 degree angle. Since W is the longest side in this triangle but not opposite Z in this larger triangle then I can say that this triangle right here must be smaller than this triangle right here.

So say that again in different way, if the longest side in this triangle is not the longest side in this triangle, then this triangle right here must be smaller. So the only side that we know and this is x, and then x must be smaller than w, and then comparing y and z we see that if this is 30 degrees and if this is 90, then you’re missing angle right here must be 60. Since 30 is smaller than 60 and 30 is the opposite w, then 60 is the opposite of y, and last z is our hypotenuse; the side that is opposite your 90 degree angle. So what do we have to do, first we decided which triangle is smaller, since we asked for the smallest, and then we say within that triangle which one has to be smaller and we worked our way through.

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