Chapter 5.4, Problem 27PPS

To determine

The solution of the statement of indirect proof.

Explanation of Solution

Given information:

Given: $XZ>YZ$

Prove: $\angle X\cancel{\cong }\angle Y$

  

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 $\angle X\cong \angle Y$ is true.

Given: $XZ>YZ$

Prove: $\angle X$ is not congruent to $\angle Y$ .

  

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

Firstly, assume that $\angle X\cong \angle Y$

Secondly, $\overline{XZ}\cong \overline{YZ}$ by the converse of the isosceles triangle Theorem.

This contradicts the given information that $XZ>YZ$ . Therefore, the assumption $\angle X\cong \angle Y$ must be false, so the original conclusion $\angle X$ is not congruent to $\angle Y$ must be true.