Students discover the relationship between the side lengths and angle measures of a triangle and apply this relationship.
After completing this tutorial, you will be able to complete the following:
The relationship between side lengths and angle measures in a triangle is a fundamental concept in Euclidean geometry. This relationship states that if are two side lengths of a triangle and are the measures of the angles opposite them, respectively, then if and only if If , then because we then have an isosceles triangle. 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 180 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 180 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. It occurs when the two propositions, taken together, yield conclusions which contradict each other. For example, the statement, "It is raining and it is not raining" is a contradiction.
Proof by contradiction, also known as reduction ad absurdum or indirect proof, is a form of argument in which a proposition is disproven by following its implications to an absurd consequence; i.e., a contradiction. In this case, a proposition is proven true by demonstrating that it is impossible for it to be false. Suppose p and q are statements, and we wish to show that p implies q. If we provided a direct proof, we would assume p to be true and show that q to also be true. However, if we assume p to be true and q to be false and use a string of implications to show that this implies that p must be false, then we have a contradiction (p is true and p is false), so it cannot be the case that p is true and q is false. Therefore, p must imply q. This powerful technique can be used in a variety of proofs and is used in this Activity Object to show the relationship between side lengths and angle measures in a triangle.
|Approximate Time||20 Minutes|
|Pre-requisite Concepts||Students should know the definitions and properties of a triangle, an isosceles triangle, and an inequality; the exterior angle and Pythagorean theorems; the meaning of proof by contradiction; and the sum of the measures of the interior angles of a triangle.|
|Type of Tutorial||Visual Proof|
|Key Vocabulary||angle, inequality, exterior angle theorem|