Reverse a Statement without Assuming Its Truth

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Write the converse: If a quadrilateral is a square, then it is a rectangle. Candidate answers: (A) If a quadrilateral is a square, then not (it is a rectangle). (B) If it is a rectangle, then a quadrilateral is a square. (C) If not (a quadrilateral is a square), then not (it is a rectangle). (D) If not (it is a rectangle), then a quadrilateral is a square..
• Independent check inputs: Write the converse: If two lines are perpendicular, then they meet at a right angle. Candidate answers: (A) If not (they meet at a right angle), then two lines are perpendicular. (B) If not (two lines are perpendicular), then not (they meet at a right angle). (C) If they meet at a right angle, then two lines are perpendicular. (D) If two lines are perpendicular, then not (they meet at a right angle)..

Learn the idea

Geometry arguments connect definitions, givens and established properties. A conclusion needs a chain of reasons; an attractive diagram may suggest a relation but does not establish every length, angle or universal claim. A converse swaps an implication’s hypothesis and conclusion. It forms a new statement that must be evaluated separately, even when a familiar geometric theorem happens to make both directions true.

Here is the method on the supplied teaching case. A converse swaps hypothesis and conclusion. It creates a new statement whose truth needs its own justification; writing the reverse does not by itself prove it.

Reversing a statement differs from negating its parts. Writing a converse is a language-and-logic step, while proving that converse needs definitions or additional reasoning rather than mere word order.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Write the converse: If an angle is a right angle, then its measure is ninety degrees. Candidate answers: (A) If an angle is a right angle, then not (its measure is ninety degrees). (B) If its measure is ninety degrees, then an angle is a right angle. (C) If not (an angle is a right angle), then not (its measure is ninety degrees). (D) If not (its measure is ninety degrees), then an angle is a right angle.. Show the mathematical reason, not only the selected answer. Expected result: If its measure is ninety degrees, then an angle is a right angle.. A converse swaps hypothesis and conclusion. It creates a new statement whose truth needs its own justification; writing the reverse does not by itself prove it.

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