GMAT Math : Geometry

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Example Questions

Example Question #2247 : Gmat Quantitative Reasoning

Lines_3

Note: Figure NOT drawn to scale.

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

Statement 1: $\angle 4$ and $\angle 5$ are complementary.

Statement 2: $OD \cdot \sqrt{2} = OE$

Possible Answers:

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.

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.

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. $\angle 5$ and $\angle 6$ are vertical from $\angle 1$ and $\angle 2$ , respectively, so $m \angle 1 = m \angle 2$ and $m \angle 2 = m \angle 6$ . $\angle 5$ and $\angle 6$ form a complementary pair, so, by definition

$m \angle 5 + m \angle 6 = 90^{\circ}$

and by substitution,

$m \angle 1 + m \angle 2 = 90^{\circ}$ .

Assume Statement 2 alone. Since $\bigtriangleup ODE$ is a right triangle whose hypotenuse is $\sqrt{2}$ times as long as a leg, it follows that $\bigtriangleup ODE$ is a 45-45-90 triangle, so $m \angle DOE = 45^{\circ }$ .

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

$m \angle 1 + m \angle 2+ m \angle 3+m \angle DOE = 180^{\circ }$

$m \angle 1 + m \angle 2+ m \angle 3+ 45^{\circ } = 180^{\circ }$

$m \angle 1 + m \angle 2+ m \angle 3 = 135^{\circ }$

But without further information, the sum of the degree measures of only $\angle 1$ and $\angle 2$ cannot be calculated.

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Example Question #22 : Lines

Lines_3

Note: Figure NOT drawn to scale.

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

Statement 1: $\angle 6$ is a $54 ^{\circ }$ angle.

Statement 2: $\angle 2 \cong \angle 3$

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 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.

Correct answer:

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

Explanation:

Statement 1 alone gives insufficient information to find the measure of $\angle 1$ .

$\angle 6$ , $\angle 1$ , and $\angle 2$ together form a $180^{\circ }$ angle; therefore,

$m \angle 1+m \angle2+ m \angle 6 = 180^{\circ }$

$m \angle 6 = 54^{\circ }$ , so by substitution,

$m \angle 1+m \angle2+ 54^{\circ } = 180^{\circ }$

$m \angle 1+m \angle2= 126^{\circ }$

But with no further information, the measure of $\angle 1$ cannot be calculated.

Statement 2 alone gives insufficient information for a similar reason. $\angle 1$ , $\angle 2$ , and $\angle 3$ together form a $180^{\circ }$ angle; therefore,

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

Since $\angle 2 \cong \angle 3$ , we can rewrite this statement as

$m \angle 1+m \angle2+ m \angle 2 = 180^{\circ }$

$m \angle 1+2 \cdot m \angle2 = 180^{\circ }$

Again, with no further information, the measure of $\angle 1$ cannot be calculated.

Assume both statements to be true. $\angle 2$ and $\angle 6$ are a pair of vertical angles, so $\angle 2 \cong \angle 6$ , and $m \angle 2 = m \angle 6 = 54^{\circ }$ . Since $\angle 2 \cong \angle 3$ , then $m \angle 3 = m \angle 2 = 54^{\circ }$ . Also,

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

By substitution,

$m \angle 1+ 54^{\circ }+ 54^{\circ } = 180^{\circ }$

$m \angle 1+108^{\circ } = 180^{\circ }$

$m \angle 1=72^{\circ }$

Report an Error

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 ,

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

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

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

Report an Error

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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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #2247 : Gmat Quantitative Reasoning

Example Question #22 : Lines

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

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