Chapter 5.4, Problem 28PPS

To determine

The solution of the statement of indirect proof.

Explanation of Solution

Given information:

Given: $\angle ABC$ is equilateral.

Prove: $\angle ABC$ is equiangular.

  

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, assume that $\angle ABC$ is not equiangular.

Given: $\angle ABC$ is equilateral.

Prove: $\angle ABC$ is equiangular.

  

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

Firstly, assume that $\angle ABC$ is not equiangular.

Secondly, $m\angle B>m\angle C$ . Then $AC>AB$ by Angle-Side Relationship in Triangles Theorem.

This contradicts the given information that $\angle ABC$ is equilateral. Therefore, the assumption that $\angle ABC$ is not equiangular must be false, so the original conclusion $\angle ABC$ is equiangular must be true.