GMAT Math : Data-Sufficiency Questions

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Example Question #141 : Data Sufficiency Questions

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above figure. Evaluate $m \angle 1+ m \angle 2$ .

Statement 1: $m \angle 3 + m \angle 4 + m \angle 5 = 131^{\circ }$

Statement 2: $m \angle 6 + m \angle 7 + m \angle 8 = 131^{\circ }$

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.

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.

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.

Explanation:

Assume Statement 1 alone. $\angle 2$ , $\angle 3$ , $\angle 4$ , and $\angle 5$ together form a straight angle, so their degree measures total $180^{\circ }$ .

$m \angle 2 + m \angle 3+ m \angle 4+m \angle 5 = 180^{\circ }$

$m \angle 2 + 131^{\circ } = 180^{\circ }$

$m \angle 2 = 49^{\circ }$

Without further information, no other angle measures, including that of $\angle 1$ , can be found.

Assume Statement 2 alone. $\angle 1$ , $\angle 6$ , $\angle 7$ , and $\angle 8$ together form a straight angle, so their degree measures total $180^{\circ }$ .

$m \angle 1 + m \angle 6+ m \angle 7+m \angle 8 = 180^{\circ }$

$m \angle 1 + 131^{\circ } = 180^{\circ }$

$m \angle 1 = 49^{\circ }$

Without further information, no other angle measures, including that of $\angle 2$ , can be found.

However, if both statements are assumed to be true, it follows from Statements 2 and 1 respectively, as seen before, that $m \angle 1 = 49^{\circ }$ and $m \angle 2 = 49^{\circ }$ , so

$m \angle 1 + m \angle 2 = 49^{\circ }+ 49^{\circ } = 98^{\circ }$ .

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Example Question #27 : Lines

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above figure. Give the measure of $\angle 1$ .

Statement 1: $m \angle 2 + m \angle 3 = 91^{\circ }$

Statement 2: $m \angle 5 + m \angle 6 = 91^{\circ }$

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.

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.

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

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements to be true. We show that the two statements provide insufficient information by exploring two scenarios:

Case 1: $m \angle 1 = m \angle 3 = 45 ^{\circ }, m \angle 2 = 46^{\circ }$

$m \angle 2 + m \angle 3 = 46 ^{\circ }+ 45^{\circ } = 91^{\circ }$

$\angle 5$ and $\angle 6$ are vertical from $\angle 1$ and $\angle 2$ , respectively, so $m \angle 5 = m \angle 1$ and $m \angle 6 = m \angle 2$ , and

$m \angle 5 + m \angle 6 =m \angle 1+ m \angle 2 = 45 ^{\circ }+ 46^{\circ } = 91^{\circ }$

Case 2: $m \angle 1 = m \angle 3 = 46 ^{\circ }, m \angle 2 = 45^{\circ }$

$m \angle 2 + m \angle 3 = 45 ^{\circ }+ 46^{\circ } = 91^{\circ }$

$m \angle 5 + m \angle 6 =m \angle 1+ m \angle 2 = 46 ^{\circ }+ 45^{\circ } = 91^{\circ }$

The conditions of both statements are met, but $\angle 1$ assumes a different value in each scenario.

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Example Question #28 : Lines

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate $m \angle 1+ m \angle 2$ .

Statement 1: $m \angle 2+ m \angle 3 = 122^{\circ }$

Statement 2: $m \angle 3+ m \angle 4 = 126^{\circ }$

Possible Answers:

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.

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 insufficient 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.

Explanation:

Assume Statement 1 alone. $\angle 1$ , $\angle 2$ , and $\angle 3$ together form a straight angle, so their measures total $180^{\circ }$ ; therefore,

$m \angle 1+m \angle2+ m \angle 3 = 180^{\circ }$

$m \angle 1+122 ^{\circ } = 180^{\circ }$

$m \angle 1=58^{\circ }$

However, without any further information, we cannot determine the sum of the measures of $\angle 1$ and $\angle 2$ .

Assume Statement 2 alone. $\angle 2$ , $\angle 3$ , and $\angle 4$ together form a straight angle, so their measures total $180^{\circ }$ ; therefore,

$m \angle2+ m \angle 3 + m \angle 4= 180^{\circ }$

$m \angle 2+126 ^{\circ } = 180^{\circ }$

$m \angle 2 =54^{\circ }$

Again, without any further information, we cannot determine the sum of the measures of $\angle 1$ and $\angle 2$ .

Assume both statements are true. Since the measures of $\angle 1$ and $\angle 2$ can be calculated from Statements 1 and 2, respectively. We can add them:

$m \angle 1+ m \angle 2 = 58^{\circ }+ 54^{\circ }=112^{\circ }$

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Example Question #29 : Lines

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate $m \angle 3+ m \angle 4$ .

Statement 1: $m \angle1 + m \angle 5 = 120^{\circ }$

Statement 2: $\bigtriangleup OPQ$ is an equilateral triangle.

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.

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 insufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. $\angle 1$ and $\angle 4$ are a pair of vertical angles, as are $\angle 3$ and $\angle 5$ . Therefore,

$m \angle 4 = m \angle 1$

$m \angle 3 = m \angle 5$

By substitution,

$m \angle 3+ m \angle 4 = m \angle 1 + m \angle 5 = 120 ^{\circ}$ .

Assume Statement 2 alone. The angles of an equilateral triangle all measure $60^{\circ }$ , so $m \angle POQ = 60 ^{\circ }$ .

$\angle 3$ , $\angle 4$ , and $\angle POQ$ together form a straight angle, so ,

$m \angle 3 + m \angle 4 + m \angle POQ= 180^{\circ }$

$m \angle 3 + m \angle 4 +60^{\circ } = 180^{\circ }$

$m \angle 3+ m \angle 4 = 120 ^{\circ}$

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Example Question #1 : Dsq: Understanding Intersecting Lines

Lines

Refer to the above figure. Jane chose one of the line segments shown in the above diagram but she will not reveal which one. Which one did she choose?

Statement 1: One of the endpoints of the line segment is .

Statement 2: The line segment includes .

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:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

If we know both statements, then we know that the segment can be either $\overline{BC}$ or $\overline{BD }$ , since each has endpoint and each includes ; we can not eliminate either, however.

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Example Question #2 : Dsq: Understanding Intersecting Lines

How many times does f(x) and g(f) intersect?

I) g(f) is a linear equation with a slope of .

II) f(x) is quadratic equation with a vertex at (-76,79) .

Possible Answers:

Either statement is sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Both statements are needed to answer the question.

Correct answer:

Neither statement is sufficient to answer the question. More information is needed.

Explanation:

When we have a linear equation and a quadratic equation there are only so many times they can intersect. They can intersect 0 times, once, or twice.

I) Gives us the slope of one equation.

II) Gives us the vertex of our quadratic equation.

If you draw a picture, it should be apparent that we don't have enough information to know exactly how many times they intersect. Our quadratic could be facing up or down, and our linear equation could go straight through both arms, or it could miss it entirely. Therfore, neither statement is sufficient.

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Example Question #3 : Dsq: Understanding Intersecting Lines

Find the 4 angles created by the two intersecting lines.

Statement 1: y=0 and x=0

Statement 2: y=2x+10 and $y=-\frac{1}{2}x+4$

Possible Answers:

$\textup{Statement 1) ALONE is sufficient, but Statement 2) ALONE is not sufficient to answer the question.}$

$\textup{Statement 2) ALONE is sufficient, but Statement 1) ALONE is not sufficient to answer the question.}$

$\textup{EACH statement ALONE is sufficient.}$

$\textup{BOTH statements TOGETHER are NOT sufficient, and additional data is needed to answer the question. }$

$\textup{BOTH statements taken TOGETHER are sufficient to answer the question, but neither statement ALONE is sufficient.}$

Correct answer:

$\textup{EACH statement ALONE is sufficient.}$

Explanation:

Statement 1: y=0 and x=0

The line y=0 is a horizontal line on the x-axis. The line x=0 is a vertical line graphed along the y-axis. The lines will create perpendicular angles, which are all 90 degrees.

Statement 2: y=2x+10 and $y=-\frac{1}{2}x+4$

These two functions are in y=mx+b form, which allows us to determine the slopes of these functions. The slopes are 2 and negative half, which are both the negative reciprocal to each other. The property of negative reciprocal slopes state that these two lines are also perpendicular to each other.

Therefore:

$\textup{EACH statement ALONE is sufficient.}$

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Example Question #4 : Dsq: Understanding Intersecting Lines

Determine the value of the four angles created by the intersecting lines.

Statement 1: Two angles are acute, and two angles are obtuse.

Statement 2: Any two non-perpendicular intersecting lines with known equations.

Possible Answers:

$\textup{EACH statement ALONE is sufficient.}$

$\textup{Statement 1) ALONE is sufficient, but Statement 2) ALONE is not sufficient to answer the question.}$

$\textup{Statement 2) ALONE is sufficient, but Statement 1) ALONE is not sufficient to answer the question.}$

$\textup{BOTH statements TOGETHER are NOT sufficient, and additional data is needed to answer the question. }$

$\textup{BOTH statements taken TOGETHER are sufficient to answer the question, but neither statement ALONE is sufficient.}$

Correct answer:

$\textup{Statement 2) ALONE is sufficient, but Statement 1) ALONE is not sufficient to answer the question.}$

Explanation:

Statement 1): Two angles are acute, and two angles are obtuse.

This statement is not necessarily true. Two intersecting lines may also be perpendicular to each other, which means that all four angles are 90 degrees.

There is not enough information to justify this statement.

Statement 2: Any two non-perpendicular intersecting lines with known equations.

This is a tricky statement.

When two functions meet, they must have an intersecting point $\textup{P}$ . Both functions $\textup{f1, f2}$ can be set equal to each other to determine that intersecting point.

Draw an imaginary line $\textup{f3}$ where the line is perpendicular to the first function and passes through the second function at some known arbitrary point $\textup{P2}$ . Point $\textup{P1}$ will need to be determined.

The equation of the third function can be determined since imaginary line $\textup{f3}$ intersects equation $\textup{f1}$ at $\textup{P2}$ , and $\textup{f1}$ is also perpendicular to $\textup{f3}$ . The slope of $\textup{f3}$ can be determined since it's the negative reciprocal of the slope of $\textup{f1}$ .

After the equation $\textup{f3}$ has been determined by using point $\textup{P2}$ and the slope of $\textup{f1}$ , the point $\textup{P1}$ can also be determined by setting the functions $\textup{f2 and f3}$ equal to each other.

Once the points $\textup{P, P1, and P2}$ have been determined, the distance formula may be used to determine the lengths from $\textup{P to P1}$ , $\textup{P to P2}$ , and $\textup{P1 to P2}$ .

The Law of Sines can then be used to determine the interior angles of the triangle bounded by $\textup{P, P1, and P2}$ . Knowing one angle at the intersection of $\textup{f1, f2}$ is sufficient to solve for all four angles by supplementary and opposite angle rules.

Therefore:

$\textup{Statement 2) ALONE is sufficient, but Statement 1) ALONE is not sufficient to answer the question.}$

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Example Question #1 : Dsq: Understanding Intersecting Lines

Parallel lines

What is the value of x?

(1) y=25

(2) Lines p and q are parallel

Possible Answers:

Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

EACH statement ALONE is sufficient to answer the question asked.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Correct answer:

BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

Explanation:

With just statement 1, there is no definitive relationship between angle y and angle x.

With just statement 2, there is a difinitive relationship between angle y and angle x, but we don't know the measure of angle y.

If you have the information from both statements 1 and 2, you can determine the measure of $\tiny \small \measuredangle x is 180-25$ , so $\tiny x=155$ .

Fortunately, this is a data sufficiency question, so you don't have to actually do the math, you just have to know that you have all the information to do the math.

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Example Question #1 : Dsq: Understanding Intersecting Lines

Parallel lines

What is the value of x?

(1) $\measuredangle z=30$

(2) $\measuredangle y=150$

Possible Answers:

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

EACH statement ALONE is sufficient to answer the question asked.

BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

Correct answer:

Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Explanation:

With just statement 1, we know $\measuredangle z=30$ , so we could determine the measure of angle y, but there is no definitive relationship between z and x nor y and x, because we don't know if lines p and q are parallel.

With just statement 2,we know $\measuredangle y=50$ , so we could determine the measure of angle z, but there is no definitive relationship between y and x nor z and x, because we don't know if lines p and q are parallel.

Even if we have the information from both statements 1 and 2, we still do not know if lines p and q are parallel, therefore there is no difinitive relationship between angle y and angle x nor angle z and angle x.

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GMAT Math : Data-Sufficiency Questions

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #141 : Data Sufficiency Questions

Example Question #27 : Lines

Example Question #28 : Lines

Example Question #29 : Lines

Example Question #1 : Dsq: Understanding Intersecting Lines

Example Question #2 : Dsq: Understanding Intersecting Lines

Example Question #3 : Dsq: Understanding Intersecting Lines

Example Question #4 : Dsq: Understanding Intersecting Lines

Example Question #1 : Dsq: Understanding Intersecting Lines

Example Question #1 : Dsq: Understanding Intersecting Lines

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