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## Example Questions

### Example Question #1 : Dsq: Calculating Whether Lines Are Parallel

You are given two lines. Are they parallel?

Statement 1: The product of their slopes is .

Statement 2: One has positive slope; one has negative slope.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

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.

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

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

Two parallel lines must have the same slope. Therefore, the product of the slopes will be the product of two real numbers of like sign, which must be positive. Each of the two statements contradicts this, so either statement alone answers the question.

### Example Question #1 : Dsq: Calculating Whether Lines Are Parallel

One line includes the points and ; a second line includes the points and . If these lines are parallel, what is the value of ?

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.

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.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

The lines are parallel, so their slopes are equal.

The slope of the first is .

The slope of the second is .

Set the two equal to each other:

If you know that , then you can easily find by substituting:

Cross-multiply and solve:

If you know that , do the same thing:

Therefore, either statement alone is sufficient to answer the question.

### Example Question #2 : Dsq: Calculating Whether Lines Are Parallel

You are given *distinct* lines and on the coordinate plane. Are they parallel, perpendicular, or neither?

Statement 1: Both lines have slope 3.

Statement 2: Line has -intercept and Line has -intercept .

**Possible Answers:**

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

Two lines can be determined to be parallel, perpendicular, or neither from their slopes.

Assume Statement 1 alone is true. Since these distinct lines have the same slope, they are parallel.

Assume Statement 2 alone is true. No information about the slopes of the lines can be determined from one single point, so Statement 2 alone is insufficient.

### Example Question #3 : Dsq: Calculating Whether Lines Are Parallel

You are given distinct lines and on the coordinate plane. Are they parallel, perpendicular, or neither?

Statement 1: Line has slope 3 and Line has slope .

Statement 2: Both lines have -intercept .

**Possible Answers:**

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.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Two lines can be determined to be parallel, perpendicular, or neither from their slopes.

Assume Statement 1 alone. Parallel lines must have the same slope, so this choice can be eliminated. The slopes of perpendicular lines must have product ; since the product of the slopes is , this choice can be eliminated as well. It can therefore be deduced that the lines are neither parallel nor perpendicular.

Assume Statement 2 alone. Since the lines have at least one point in common, they are not parallel, but this is the only choice that can be eliminated.

### Example Question #4 : Dsq: Calculating Whether Lines Are Parallel

You are given distinct lines and on the coordinate plane. Are they parallel, perpendicular, or neither?

Statement 1: Line has slope 3 and Line has slope .

Statement 2: Line has -intercept and Line has -intercept .

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Two lines are parallel if and only if they have the same slope, and perpendicular if and only if their slopes have product .

Assume Statement 1 alone. Since the product of the slopes is , the lines are perpendicular.

Statement 2 alone is unhelpful, since no information about the slope of a line can be determined from only one point.

### Example Question #5 : Dsq: Calculating Whether Lines Are Parallel

You are given two *distinct* lines, and on the coordinate plane. Are they parallel lines, perpendicular lines, or neither of these?

Statement 1: The absolute value of the slope of Line is 1.

Statement 2: The absolute value of the slope of Line is 1.

**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.

Assume both statements are true. Then three things are possible:

Case 1: Both lines will have slope 1, or

Case 2: Both lines will have slope

In either case, since the lines have the same slope, they are parallel.

Case 3: One line has slope 1 and one has slope

In this case the lines are perpendicular.

The two statements therefore provide insufficient information.

### Example Question #6 : Dsq: Calculating Whether Lines Are Parallel

Statement 1: Line has -intercept and line has -intercept .

Statement 1: Line has -intercept and line has -intercept .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

The answer to the question depends on the slopes of the lines—parallel lines have the same slope, and perpendicular lines have slopes that have product .

From Statement 1 alone, we only know one point of each line, so no information about their slopes can be determined; the same holds for Statement 2.

Assume both statements are true. Then we know two points of each line—specifically, both intercepts—from which we can determine the slope of each by way of the slope formula. After doing so, we can use the slopes to answer the question.

### Example Question #7 : Dsq: Calculating Whether Lines Are Parallel

You are given two *distinct* lines, Line and Line , on the coordinate plane. Neither line is horizontal or vertical. Are they parallel lines, perpendicular lines, or neither of these?

Statement 1: The product of the slopes of the two lines is .

Statement 2: The absolute value of the slope of Line is .

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

The question can be answered by finding and comparing the slopes of the lines. The lines are parallel if and only if they have the same slope, and perpendicular if and only if the product of the slopes is .

Statement 1 alone does not answer the question. Two lines with slope 1 are parallel, and a line with slope 2 and a line with slope are not, but in both cases, the product of the slopes is 1.

Statement 2 alone gives that Line has slope 1 or , but nothing is given about the slope of Line .

Now, assume both statements are true. From Statement 2, has slope 1 or . From Statement 1, the product of the slopes is 1; if the slope of is 1, then the slope of is , and if the slope of is , then the slope of is . Therefore, if both statements are true, the lines have the same slope, making them parallel.

### Example Question #8 : Dsq: Calculating Whether Lines Are Parallel

Statement 1: The product of the slopes of the two lines is .

Statement 2: The slope of Line is .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

The answer to the question depends on the slopes of the lines - parallel lines have the same slope, and perpendicular lines have slopes that have product .

Statement 1 alone only eliminates the possiblity of the lines being perpendicular, since the product of the slope is not . Two lines with slope 3 are parallel, and one line with slope 1 and one with slope 9 are neither parallel nor perpendicular; both pairs of lines satisfy Statement 1, but only the first pair is parallel. Therefore, Statement 1 only establishes that they are not perpendicular.

From Statement 2, only the slope of is given; without the slope of , the question cannot be answered.

Assume both statements to be true. Then since Line has slope and the product of the slopes is 9, The slope of Line is . Therefore, both lines have slope , and the lines are parallel.

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