Chapter 5.4, Problem 41HP

To determine

The statement of an indirect proof.

Explanation of Solution

Given information:

Write a statement that can be proven using indirect proof. Include the indirect proof of the statement.

Calculation:

In an indirect proof or proof by contradiction, assume that the conclusion is false. By showing the assumption to be logically impossible, prove the assumption is false and the original conclusion is true. For this problem, consider choosing something like a definition or a theorem that can be easily proven true by contradiction.

Given: $\angle ABC$ ; $AB\ne BC$ ; $BC\ne AC$ ; $AB\ne AC$

Prove: $\angle ABC$ is scalene.

  

Following steps are used for the indirect proof are as shown below:

Step $1:$ Assume that $\angle ABC$ is not scalene.

Case 1: $\angle ABC$ is isosceles.

Step $2:$ If $\angle ABC$ is isosceles, then $AB=BC,BC=AC,$ or $AB=AC$ .

Step $3$ : This contradicts the given information, so $\angle ABC$ is not isosceles.

Case $2:$ $\angle ABC$ is equilateral.

In order for a triangle to be a equilateral, it must also be isosceles, and case $1$ proved that $\angle ABC$ is not isosceles. Thus, $\angle ABC$ is not equilateral. Therefore, $\angle ABC$ is scalene.