By now you probably realize that the relationships between the angles and the sides of a triangle can reveal additional information about the triangle. The problems on the previous page gave you information about all the angles or all the sides of the triangle. While this is the ideal situation, you may not always have this much information when asked to solve a problem. The video below demonstrates what to do when you have less information than you need to apply the Triangle Inequality Theorem.
As you watch this video, use the study guide to follow along if you'd like. Click the button below to download the study guide.
If you remember from earlier in this module, the triangle inequality theorem states that any side of a triangle must be shorter than the sum of the other two sides. For example, if I gave you the side lengths 2, 3, and 6, you couldn't make a triangle because the side lengths 2 and 3 just aren't long enough to reach each other since the side length of 6 has them spread too far apart. From this theorem, it follows that the longest side of a triangle is across from, or opposite of, the largest angle, and the shortest side is opposite of the smallest angle.
With this information let's consider the following two problems. In problem #1 we are told that in triangle ABC, angle B is forty-two degrees, and angle C is seventy-one degrees. We should be able to identify the longest and shortest side from this information. Before calculating anything, I recommend drawing a representative diagram. It doesn't need to be drawn accurately to the side lengths and angle measures, but the positions of each vertex will be important… With a diagram of ABC drawn, we can find the missing angle measure if we remember the sum of the angles must be 180 degrees… With the measure of angle A equal to sixty-seven degrees, angle B equal to forty-two degrees, and angle C equal to seventy-one degrees, we can determine the longest side is across from angle C, and the shortest side is across from angle B… Side length AB is the longest side and side length AC is the shortest. We don't know the actual lengths of these triangles, and it will be a while before we learn methods to calculate them since this triangle is not a right triangle.
Now try problem #2 on your own. Be sure to begin with a representative diagram, and fill in the diagram with the given information. Resume playback of the video in a moment to check your answers… Angle E is the smallest since it is opposite of the shortest side, and angle F is the largest since it is opposite of the longest side.
I hope this video helps you better understand the Triangle Inequality Theorem, and gives you an idea of how it can be extended for use in other applications. Be sure to re-watch this video, and practice problems like this on your own. Good Luck!
