Geometry worksheet featuring 12 conjectures to disprove using counterexamples.

Let's go through each conjecture and provide a counterexample that disproves it. A counterexample is a specific case that shows the conjecture is false.

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1) No triangles have two sides of the same length.

Counterexample:
An isosceles triangle has two sides of equal length.
For example: a triangle with side lengths 5, 5, and 8.
This contradicts the claim that *no* triangles have two equal sides.

✔ Disproven by an isosceles triangle.

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2) No women have been elected U.S. senators.

Counterexample:
Kamala Harris was elected as a U.S. Senator from California in 2016.
Also, Barbara Mikulski, Dianne Feinstein, and many others have served as U.S. Senators.

✔ Disproven by any female U.S. Senator (e.g., Kamala Harris).

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3) All basketball players are more than 6 feet tall.

Counterexample:
Muggsy Bogues, a professional NBA player, was only 5 feet 3 inches tall.
He played for the Charlotte Hornets and is one of the shortest players in NBA history.

✔ Disproven by Muggsy Bogues.

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4) If you live in Texas, then you live in Houston.

Counterexample:
Someone who lives in Austin, Texas lives in Texas but not in Houston.
Houston is just one city in Texas.

✔ Disproven by someone living in Austin or Dallas.

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5) If it is a cell phone, then it has a touch screen.

Counterexample:
A flip phone like the Motorola Razr (older models) or some basic phones use buttons instead of touch screens.
Even today, some feature phones don’t have touch screens.

✔ Disproven by a non-touchscreen cell phone (e.g., a flip phone).

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6) If x is any number, then x² > 2.

Counterexample:
Let $ x = 1 $. Then $ x^2 = 1 $, which is not greater than 2.
Also, $ x = 0 \Rightarrow x^2 = 0 $, still not > 2.

✔ Disproven by $ x = 1 $ (or $ x = 0 $).

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7) If $ x \geq 0 $, then $ x + 2 = 7 $.

Counterexample:
Let $ x = 0 $. Then $ x + 2 = 2 $, which is not equal to 7.
Or $ x = 1 \Rightarrow x+2 = 3 $, still not 7.

✔ Disproven by $ x = 0 $.

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8) If $ x^2 = 16 $, then $ x = 4 $.

Counterexample:
$ x = -4 $ also satisfies $ x^2 = 16 $, since $ (-4)^2 = 16 $.
But $ -4 \neq 4 $.

✔ Disproven by $ x = -4 $.

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9) All figures with four sides of equal length are squares.

Counterexample:
A rhombus has four equal sides, but its angles are not necessarily 90°.
For example: a rhombus with angles of 60° and 120° is not a square.

✔ Disproven by a rhombus that is not a square.

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10) The square of a number is larger than the number.

Counterexample:
Let $ x = 1 $. Then $ x^2 = 1 $, which is not larger than 1.
Also, $ x = 0 \Rightarrow x^2 = 0 $, same value.
Even $ x = 0.5 \Rightarrow x^2 = 0.25 < 0.5 $.

✔ Disproven by $ x = 1 $ (or $ x = 0.5 $).

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11) If a number is prime, then it is an odd number.

Counterexample:
The number 2 is prime, but it is even, not odd.

✔ Disproven by $ x = 2 $.

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Summary of Counterexamples:

| Problem | Counterexample |
|--------|----------------|
| 1 | Isosceles triangle (two equal sides) |
| 2 | Kamala Harris (female U.S. Senator) |
| 3 | Muggsy Bogues (NBA player under 6 ft) |
| 4 | Person living in Austin, Texas |
| 5 | Flip phone without touchscreen |
| 6 | $ x = 1 $, $ x^2 = 1 \not> 2 $ |
| 7 | $ x = 0 $, $ x + 2 = 2 \ne 7 $ |
| 8 | $ x = -4 $, $ x^2 = 16 $, but $ x \ne 4 $ |
| 9 | Rhombus (four equal sides, not a square) |
| 10 | $ x = 1 $, $ x^2 = 1 \not> 1 $ |
| 11 | $ x = 2 $, prime but even |

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✔ All conjectures are false, and we’ve provided one valid counterexample for each.

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