# GMAT Math : Lines

## Example Questions

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

Data Sufficiency Question

Is Line A perpendicular to the following line?

Statement 1: The slope of Line A is 3.

Statement 2: Line A passes through the point (2,3).

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

Each statement alone is sufficient.

Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

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

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

Explanation:

To determine if two lines are perpendicular, only the slope needs to be considered. The slopes of perpendicular lines are the negative reciprocals of each other. Knowing a single point on the line is not sufficient, as an infinite number of lines can pass through and individual point.

### Example Question #1 : Geometry

Refer to the above figure. True or false:

Statement 1:  is equilateral.

Statement 2: Line  bisects .

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.

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.

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

Explanation:

Statement 1 alone establishes nothing about the angle  makes with , as it is not part of the triangle. Statement 2 alone only establishes that .

Assume both statements are true. Then  is an altitude of an equilateral triangle, making it - and  - perpendicular with the base  - and .

### Example Question #3 : Geometry

Statement 1:

Refer to the above figure. Are the lines perpendicular?

Statement 1:

Statement 2:

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.

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.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. The measure of one of the angles formed is

degrees.

Assume Statement 2 alone.

By substituting  for , one angle measure becomes

The marked angles are a linear pair and thus their angle measures add up to 180 degrees; therefore, we can set up an equation:

or

yields illegal angle measures - for example,

yields angle measures  for both angles; the angles are right and the lines are perpendicular.

### Example Question #4 : Geometry

Refer to the above figure.

True or false:

Statement 1:

Statement 2:

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.

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 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.

Explanation:

Statement 1 alone establishes by definition that , but does not establish any relationship between  and .

By Statement 2 alone, since alternating interior angles are congruent, , but no conclusion can be drawn about the relationship of , since the actual measures of the angles are not given.

Assume both statements are true. By Statement 2,  and  are corresponding angles formed by a transversal across parallel lines, so  is not a right angle, so .

### Example Question #5 : Geometry

Refer to the above figure. True or false:

Statement 1:

Statement 2: Line  bisects .

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.

EITHER 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.

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

Explanation:

Assume Statement 1 alone. Then, as a consequence of congruence,  and  are congruent. They form a linear pair of angles, so they are also supplementary. Two angles that are both congruent and supplementary must be right angles, so .

Assume Statement 2 alone. Then , but without any other information about the angles that  or  make with , it cannot be determined whether  or not.

### Example Question #1 : Geometry

The equations of two lines are:

Are these lines perpendicular?

Statement 1:

Statement 2:

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.

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.

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.

Explanation:

The lines of the two equations must have slopes that are the opposites of each others reciprocals.

Write each equation in slope-intercept form:

As can be seen, knowing the value of  is necessary and sufficient to answer the question. The value of  is irrelevant.

The answer is that Statement 1 alone is sufficient to answer the question, but Statement 2 alone is not sufficient to answer the question.

### Example Question #1 : Lines

Note: Figure NOT drawn to scale.

Evaluate .

Statement 1:

Statement 2:

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.

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.

Explanation:

Even with both statements,  cannot be determined because the length of is missing.

For example, we can have  and , making ; or, we can have   and , making . Neither scenario violates the conditions given.

### Example Question #8 : Geometry

, and  are distinct points.

True or false:  and  are opposite rays.

Statement 1:  is on

Statement 2:  is on

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.

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

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Both statements are equivalent, as both are equivalent to stating that , and  are collinear. Therefore, it suffices to determine whether the fact that the points are collinear is sufficient to answer the question.

In both of the above figures,  , and  are collinear, so the conditions of both statements are met. But in the top figure,  and  are the same ray, since  is on ; in the bottom figure, since  and  are on opposite sides of  and  are opposite rays.

### Example Question #2 : Geometry

, and  are distinct points.

True or false:  and  are opposite rays.

Statement 1: .

Statement 2:  is the midpoint of .

BOTH statements TOGETHER are insufficient 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.

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.

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

Explanation:

We show Statement 1 alone is insufficient to determine whether the two rays are the same by looking at the figures below. In the first figure,  is the midpoint of .

In both figures, . But only in the second figure,  and  are on the opposite side of the line from , so only in the second figure,  and  are opposite rays.

Assume Statement 2 alone. If  is the midpoint of , then, as seen in the top figure,  is on . Therefore,  and  are the same ray, not opposite rays.

### Example Question #10 : Geometry

, and  are distinct points.

True or false:  and  are opposite rays.

Statement 1:

Statement 2:

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.

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.

Explanation:

Statement 1 alone does not answer the question.

Case 1: Examine the figure below.

,

thereby meeting the condition of Statement 1.

Also,  and  are opposite rays, since  and  are on opposite sides of the same line from .

Case 2: Suppose , and  are noncollinear.

The three points are vertices of a triangle, and by the Triangle Inequality Theorem,

.

Furthermore,  and  are not part of the same line and are not opposite rays.

Now assume Statement 2 alone. As can be seen in the diagram above, if  and  are opposite rays, then by segment addition, , making Statement 2 false. Contrapositively, if Statement 2 holds, and , then  and  are not opposite rays.

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