Chapter 7.1, Problem 64SPR

To determine

To find: The counterexample for the given statement.

Answer to Problem 64SPR

The counterexample is “the lines $p$ and $m$ are cut by transversal $t$ so that consecutive interior angles are congruent, then the lines $p$ and $m$ are not parallel and $t$ is perpendicular to both lines.”

Explanation of Solution

Given:

The given statement is “if lines $p$ and $m$ are cut by transversal $t$ so that consecutive interior angles are congruent, then lines $p$ and $m$ are parallel and $t$ is perpendicular to both lines.”

Calculation:

To find the counterexample for a conditional, one should keep the hypothesis true and falsify the conclusion.

Step 1: Identify the hypothesis and the conclusion of the true conditional.

Hypothesis: Lines $p$ and $m$ are cut by transversal $t$ so that consecutive interior angles are congruent;

Conclusion: Lines $p$ and $m$ are parallel and $t$ is perpendicular to both lines.

Step 2: Find the counterexample as follows.

Hypothesis: Lines $p$ and $m$ are cut by transversal $t$ so that consecutive interior angles are congruent;

Conclusion: Lines $p$ and $m$ are not parallel and $t$ is perpendicular to both lines.

  

Therefore, the counterexample is “the lines $p$ and $m$ are cut by transversal $t$ so that consecutive interior angles are congruent, then the lines $p$ and $m$ are not parallel and $t$ is perpendicular to both lines.”