Triangles: The Side-Angle Relationship
Move vertex A until side AB is the longest side. Which angle is the largest? Move vertex A until side AC is the longest side. Which angle is the la. So, the smallest angle is opposite the shortest side. Let's look at another one: What is the relationship between side B and angle y? 40 + 25 + y = => 65 + y. Students will explore the relationship between side lengths and angle measures for one triangle.
Now, we will look at an inequality that involves exterior angles. Exterior Angle Inequality Theorem The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles.
For this theorem, we only have two inequalities since we are just comparing an exterior angle to the two remote interior angles of a triangle. Let's take a look at what this theorem means in terms of the illustration we have below. 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.
Geometry: Triangle Inequality and Angle-Side Relationship
Exercise 1 In the figure below, what range of length is possible for the third side, x, to be. 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.Types of Angles and Angle Relationships
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.
Inequalities and Relationship in a Triangle | Wyzant Resources
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.
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.
Inequalities and Relationships Within a Triangle
The exterior angle we will focus on is? We have been given that? Know the relationship between the interior angles of a triangle and corresponding side lengths.
Apply the relationship between the interior angles of a triangle and the sides opposite them.
Everything You'll Have Covered The relationship between side lengths and angle measures in a triangle is a fundamental concept in Euclidean geometry. In this Activity Object, the proof of this intuitive concept relies on the exterior angle theorem.
Recall that an exterior angle of a triangle is an angle between one side of a triangle and the extension of the adjacent side. Notice that the sum of the angle measures of an exterior angle and the adjacent interior angle of the triangle is degrees because together they form a straight angle. Since we also know that the sum of the measures of the interior angles of a triangle is degrees, this means that we have the following theorem.
Exterior angle theorem-The measure of an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles of the same triangle. The proof also relies on a proof technique known as proof by contradiction. In mathematics, a contradiction consists of a logical incompatibility between two or more propositions.