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## Example Questions

### Example Question #231 : Plane Geometry

You are given triangles and , with and . Which of these statements, along with what you are given, is* not *enough to prove that ?

I) and have the same perimeter

II)

III)

**Possible Answers:**

None of these statements is enough to prove congruence.

Statement I only

Any of the three statements is enough to prove congruence.

Statement III only

Statement II only

**Correct answer:**

Statement III only

If and have the same perimeter, , and , it follows that . The three triangles have the same sidelengths, setting the conditions for the Side-Side-Side Congruence Postulate.

If , then, since the sum of the degree measures of both triangles is the same (180 degrees), it follows that . Since and are congruent included angles of congruent sides, this sets the conditions for the SAS Congruence Postulate.

In both of the above cases, it follows that .

However, similarly to the previous situation, if , then it follows that , meaning that we have congruent sides and congruent *nonincluded *angles. However, this is not sufficient to prove congruence.

"Statement III" is the correct response.

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