GMAT Math : GMAT Quantitative Reasoning

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Example Questions

Example Question #23 : Lines

Lines_4

Note: You may assume that and are not parallel lines, but you may not assume that and are parallel lines unless it is specifically stated.

Refer to the above diagram. Is the sum of the measures of $\angle ADC$ and $\angle DCB$ less than, equal to, or greater than $180^{\circ }$ ?

Statement 1: There exists a point such that lies on $\overline{ZD}$ and lies on $\overline{ZC}$ .

Statement 2: Quadrilateral ABCD is not a trapezoid.

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 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.

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

Explanation:

Assume Statement 1 alone. Since $\overline{ZD}$ exists and includes , $\overleftrightarrow{ZD}$ and $\overleftrightarrow{DA}$ are one and the same—and this is . Similarly, $\overleftrightarrow{ZC}$ is . This means that and have a point of intersection, which is . Since falls between and and falls between and , the lines intersect on the side of that includes points and . By Euclid's Fifth Postulate, the sum of the measures of $\angle ADC$ and $\angle DCB$ is less than $180^{\circ }$ .

Assume Statement 2 alone. Since it is given that $t \nparallel u$ , the other two sides, $\overline{AD}$ and $\overline{BC}$ are parallel if and only if Quadrilateral ABCD is a trapezoid, which it is not. Therefore, $\overline{AD}$ and $\overline{BC}$ are not parallel, and the sum of the degree measures of same-side interior angles $\angle ADC$ and $\angle DCB$ is not equal to $180^{\circ }$ . However, without further information, it is impossible to determine whether the sum of the measures is less than or greater than $180^{\circ }$ .

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Example Question #135 : Data Sufficiency Questions

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above figure. Evaluate $m \angle 3$ .

Statement 1: $\overline{BO} \cong \overline{BC}$

Statement 2: $m \angle 5 = 42^{\circ }$

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.

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.

Explanation:

Assume Statement 1 alone. $\overline{BO}$ and $\overline{BC}$ are congruent legs of right triangle $\bigtriangleup OBC$ , so their acute angles, one of which is $\angle BOC$ , measure $45^{\circ }$ . $\angle BOC$ and $\angle 3$ form a pair of vertical, and consequently, congruent, angles, so $m \angle 3 = 45^{\circ }$ .

Statement 2 alone gives insufficient information, as $\angle 3$ and $\angle 5$ has no particular relationship that would lead to an arithmetic relationship between their angle measures.

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Example Question #25 : Lines

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above diagram. What is the measure of $\angle 1$ ?

Statement 1: $\bigtriangleup OPQ$ is an equilateral triangle.

Statement 2: $m \angle 3 = 65^{\circ }$

Possible Answers:

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.

Correct answer:

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

Explanation:

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

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

Assume Statement 1 alone. The angles of an equilateral triangle all measure $60^{\circ }$ , so $m \angle POQ = 60 ^{\circ }$ ; $\angle POQ$ and $\angle 2$ form a pair of vertical angles, so they are congruent, and consequently, $m \angle 2 = 60^{\circ }$ . Therefore,

$m \angle 1+ 60^{\circ } + m \angle 3 = 180^{\circ }$

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

But with no further information, $m \angle 1$ cannot be calculated.

Assume Statement 2 alone. It follows that

$m \angle 1+m \angle2+ 65^{\circ }= 180^{\circ }$

$m \angle 1+m \angle2 =115^{\circ }$

Again, with no further information, $m \angle 1$ cannot be calculated.

Assume both statements to be true. $m \angle 2 = 60^{\circ }$ as a result of Statement 1, and $m \angle 3 = 65^{\circ }$ from Statement 2, so

$m \angle 1+60^{\circ } + 65^{\circ }= 180^{\circ }$

$m \angle 1+125^{\circ }= 180^{\circ }$

$m \angle 1=55^{\circ }$

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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 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.

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.

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:

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

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.

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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GMAT Math : GMAT Quantitative Reasoning

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #23 : Lines

Example Question #135 : Data Sufficiency Questions

Example Question #25 : Lines

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

All GMAT Math Resources

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