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## Example Questions

### Example Question #1 : Dsq: Calculating Whether Acute / Obtuse Triangles Are Congruent

and

Is it true that ?

1)

2)

**Possible Answers:**

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is not sufficient.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is not sufficient.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

If only one of the statements is known to be true, the only congruent pairs that are known between the triangles comprise two sides and a *non-included* angle; this information cannot prove congruence between the triangles. If both are known to be true, however, they, along with either of the given side congruences, set up the conditions for the Angle-Angle-Side Theorem, and the triangles can be proved congruent.

The answer is that both statements together are sufficient to answer the question, but not either alone.

### Example Question #2 : Dsq: Calculating Whether Acute / Obtuse Triangles Are Congruent

You are given two triangles and ; with and . Which side is longer, or ?

Statement 1:

Statement 2: and are both right angles.

**Possible Answers:**

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is not sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is not sufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

We are given two triangles with two side congruences between them. If we compare their included angles (the angles that they form), the angle that is of greater measure will have the longer side opposite it. This is known as the *Hinge** Theorem*.

The first statement says explicitly that the first included angle, , has greater measure than the second, , so the side opposite , , has greater measure than .

The second statement is not so explicit. But if is a right angle, must be acute, and if is right, then , which again proves that .

The answer is that either statement alone is sufficient to answer the question.

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