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## Example Questions

### Example Question #1 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Is the triangle isosceles?

Statement 1: The triangle has vertices A(1,5), B(4,2), and C(5,6).

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statements 1 and 2 TOGETHER are NOT sufficient.

Statement 1 ALONE is sufficient, but statement 2 is not sufficient.

EACH statement ALONE is sufficient.

Statement 2 ALONE is sufficient, but statement 1 is not sufficient.

**Correct answer:**

EACH statement ALONE is sufficient.

For a triangle to be isosceles, two of the sides must be equal. To determine wheter this is true, we must have the three side lengths. Statement 2 gives us those three side lengths. However, Statement 1 also gives us all of the information we need by giving us the three vertices. By using the distance formula, we can easily get the three triangle sides from the vertices. Therefore both statements alone are sufficient.

### Example Question #2 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Note: Figure NOT drawn to scale.

The above shows a triangle inscribed inside a rectangle . is isosceles?

Statement 1: is the midpoint of .

Statement 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 to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

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

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

We show Statement 1 alone is sufficient:

If is the midpoint of , then . Opposite sides of a rectangle are congruent, so ; all angles of a rectangle, being right angles, are congruent, so . This sets up the conditions for the Side-Angle-Side Theorem, and . Consequently, , and is isosceles.

Now, we show Statement 2 alone is sufficient:

If , and are congruent, then and , being complements of congruent angles, are congruent themselves. By the Isosceles Triangle Theorem, is isosceles.

### Example Question #3 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Which side of is the longest?

Statement 1: is an obtuse angle.

Statement 2: and are both acute angles.

**Possible Answers:**

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.

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

EITHER statement ALONE is sufficient to answer the question.

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

**Correct answer:**

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

If we only know that two interior angles of a triangle are acute, we cannot deduce the measure of the third, or even if it is obtuse or right; therefore, Statement 2 alone does not help us.

If we know that is an obtuse angle, however, we can deduce that and are both acute angles, since at least two interior angles of a triangle are acute. Therefore, we can deduce that has the greatest measure, and that its opposite side, ,** ** is the longest.

### Example Question #4 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Is isosceles?

Statement 1:

Statement 2:

**Possible Answers:**

Statement 2 ALONE is sufficient to answer the question, but Statement 1 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.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Statement 1 alone does not tell us anything unless we know the relative lengths of the sides of ; Statement 2 only gives us information about another triangle.

Suppose we assume both statements. Then by similarity,

.

Since , then

, or

.

This makes isosceles.

### Example Question #5 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Which of the three sides of is the longest?

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

The longest side of a triangle is opposite the angle of greatest measure.

From Statement 1 alone, we can find two possible scenarios with different answers:

Case 1:

Case 2:

In both cases, , but in Case 1, is the longest side, and in Case 2, is the longest side.

From Statement 2 alone, however, we know that , so is obtuse and the other two angles are acute. That makes the longest side.

### Example Question #6 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

True or false: is scalene.

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

Assume both statements are true.

By definition, a scalene triangle has three noncongruent sides. Sides opposite noncongruent angles of a triangle are noncongruent, so as a consequence of Statement 1, . Statement 2 alone establishes that . However, the two statements together do not establish whether or not , so it is not clear whether is scalene or isosceles.

### Example Question #7 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

True or false: is scalene.

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

By definition, a scalene triangle has three noncongruent sides.

Statement 1 alone states that two sides are noncongruent, but no information is given about whether or not third side is congruent to either of the other sides.

Assume Statement 2 alone. In a triangle, sides opposite congruent angles are congruent, so it follows that . The triangle cannot be scalene.

### Example Question #8 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

True or false: is scalene.

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

By definition, a scalene triangle has three noncongruent sides.

If , then and , and the triangle is scalene.

If , then and , but , so the triangle is not scalene.

The two statements together are insufficient.

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