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## Example Questions

### Example Question #1 : How To Find If Right Triangles Are Congruent

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

I)

II)

III) and have the same area.

**Possible Answers:**

Statement II only

Statement III only

Any of the three statements is enough to prove congruence.

Statement I only

None of the three statements is enough to prove congruence.

**Correct answer:**

Any of the three statements is enough to prove congruence.

, and the right angles are and , so we have two right triangles with congruent legs.

If we also know that , then the hypotenuses of the right triangles are also congruent, and this sets up the conditions of the Hypotenuse-Leg Theorem.

If we also know that , then, along with the fact that (both being right angles) and nonincluded sides , the conditions of the Angle-Angle-Side Theorem are set up.

If we also know and have the same area, we can demonstrate that the other legs are congruent. The area of a right triangle is half the product of its legs, and since we have the same areas,

Since ,

The legs and the included angles (the right angles) are congruent, thus setting up the conditions for the Angle-Side-Angle Postulate.

In all three cases, congruence follows, so the correct response is "Any of the three statements is enough to prove congruence."

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