Use Transitivity to Connect Equal Segments

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Given AB=BC, BC=CD, and AB=5 cm, find CD. Candidate answers: (A) 10 cm (B) 4 cm (C) 5 cm (D) 6 cm.
• Independent check inputs: Given AB=BC, BC=CD, and AB=6 cm, find CD. Candidate answers: (A) 12 cm (B) 5 cm (C) 6 cm (D) 7 cm.

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. If a first segment equals a second and that second equals a third, transitivity establishes equality of the first and third. This is a logical chain rather than a visual estimate.

Here is the method on the supplied teaching case. Transitivity gives AB=CD through their shared equality to BC. Since AB=5 cm, CD=5 cm. The givens establish this value; no measurement from a sketch is needed.

Only supplied or proved equalities can enter the chain. A labeled sketch may help organize corresponding segments, but the proof needs the equalities and their transitive reason stated explicitly.

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: Given AB=BC, BC=CD, and AB=12 cm, find CD. Candidate answers: (A) 11 cm (B) 12 cm (C) 13 cm (D) 24 cm. Show the mathematical reason, not only the selected answer. Expected result: 12 cm. Transitivity gives AB=CD through their shared equality to BC. Since AB=12 cm, CD=12 cm. The givens establish this value; no measurement from a sketch is needed.

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