Triangle Inequalities: Sides and Angles
Recall that in a scalene triangle, all the sides have different lengths and all the interior angles have different measures. In such a triangle, the shortest side is. Triangle Inequality Theorem and Angle-Side Relationships in triangles, Converse of the Triangle Inequality Theorem, Angle-Side Relationship for triangles. In this section, we will learn about the inequalities and relationships within a triangle that reveal information about triangle sides and angles. First, let's take a.
So you have the side of length Now the angle is essentially 0, this angle that we care about. So this side is length 6.
And so what is the distance between this point and this point? And that distance is length x. So in the degenerate case, this length right over here is x. We know that 6 plus x is going to be equal to So in this degenerate case, x is going to be equal to 4. So if you want this to be a real triangle, at x equals 4 you've got these points as close as possible. It's degenerated into a line, into a line segment.
If you want this to be a triangle, x has to be greater than 4. Now let's think about it the other way. How large can x be?Geometry - Triangle Inequalities for sides
Well to think about larger and larger x's, we need to make this angle bigger. So let's try to do that.
So let's draw my 10 side again. So this is my 10 side.
I'm going to make that angle bigger and bigger. So now let me take my 6 side and put it like that. And so now our angle is getting bigger and bigger and bigger. It's approaching degrees. At degrees, our triangle once again will be turned into a line segment. It'll become a degenerate triangle. So let me draw the side of length x, try to draw it straight. So we're trying to maximize the distance between that point and that point.
So this is side of length x and let's go all the way to the degenerate case. In the degenerate case, at degrees, the side of length 6 forms a straight line with the side of length And this is how you can get this point and that point as far apart as possible. Well, in this situation, what is the distance between that point and that point, which is the distance which is going to be our x? Well in this situation, x is going to be 6 plus 10 is If x is 16, we have a degenerate triangle.
If we don't want a degenerate triangle, if we want to have two dimensions to the triangle, then x is going to have to be less than Now the whole principle that we're working on right over here is called the triangle inequality theorem and it's a pretty basic idea. That any one side of a triangle has to be less, if you don't want a degenerate triangle, than the sum of the other two sides. So length of a side has to be less than the sum of the lengths of other two sides. By the Exterior Angle Inequality Theorem, we have the following two pieces of information: We will use this theorem again in a proof at the end of this section.
Now, let's study some angle-side triangle relationships. Relationships of a Triangle The placement of a triangle's sides and angles is very important. We have worked with triangles extensively, but one important detail we have probably overlooked is the relationship between a triangle's sides and angles.
These angle-side relationships characterize all triangles, so it will be important to understand these relationships in order to enrich our knowledge of triangles. Angle-Side Relationships If one side of a triangle is longer than another side, then the angle opposite the longer side will have a greater degree measure than the angle opposite the shorter side.
If one angle of a triangle has a greater degree measure than another angle, then the side opposite the greater angle will be longer than the side opposite the smaller angle.
In short, we just need to understand that the larger sides of a triangle lie opposite of larger angles, and that the smaller sides of a triangle lie opposite of smaller angles.
Let's look at the figures below to organize this concept pictorially. Since segment BC is the longest side, the angle opposite of this side,? A, is has the largest measure in? C, tells us that segment AB is the smallest side of? Now, we can work on some exercises to utilize our knowledge of the inequalities and relationships within a triangle. Exercise 1 In the figure below, what range of length is possible for the third side, x, to be.
Inequalities and Relationships Within a Triangle
When considering the side lengths of a triangle, we want to use the Triangle Inequality Theorem. Recall, that this theorem requires us to compare the length of one side of the triangle, with the sum of the other two sides.
The sum of the two sides should always be greater than the length of one side in order for the figure to be a triangle. Let's write our first inequality. So, we know that x must be greater than 3. Let's see if our next inequality helps us narrow down the possible values of x.
This inequality has shown us that the value of x can be no more than Let's work out our final inequality. This final inequality does not help us narrow down our options because we were already aware of the fact that x had to be greater than 3.
Moreover, side lengths of triangles cannot be negative, so we can disregard this inequality. Combining our first two inequalities yields So, using the Triangle Inequality Theorem shows us that x must have a length between 3 and Exercise 2 List the angles in order from least to greatest measure. For this exercise, we want to use the information we know about angle-side relationships.
Since all side lengths have been given to us, we just need to order them in order from least to greatest, and then look at the angles opposite those sides. This means that the angles opposite those sides will be ordered from least to greatest. So, in order from least to greatest angle measure, we have? Exercise 3 Which side of the triangle below is the smallest?
In order to find out which side of the triangle is the smallest, we must first figure out which angle of the triangle is the smallest because the smallest side will be opposite the smallest angle. So, we must use the Triangle Angle Sum Theorem to figure out the measure of the missing angle.
Triangle Inequality Theorem and Side Angle Relationship in Triangle
V has the smallest measure, we know that the side opposite this angle has the smallest length. The corresponding side is segment DE, so DE is the shortest side of? While it may not immediately be clear that there are two exterior angles given in the diagram, we must notice them in order to establish a relationship between the two triangles' angles. The exterior angle we will focus on is?
We have been given that? KMJ are congruent, which means that the measure of their angles is equal. We also know that the measure of? JKM Is greater than either of the remote interior angles of?
Thus, we know that the measure of?