GMAT Math : Geometry

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Example Questions

Example Question #1 : Dsq: Understanding Rays

, , and are distinct points.

True or false: $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray.

Statement 1: $AB = 2 \cdot AC$ .

Statement 2: is the midpoint of $\overline{AB}$ .

Possible Answers:

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.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

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:

Rays

In both figures, $AB = 2 \cdot AC$ , but only in the first figure, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray.

Assume Statement 2 alone. If is the midpoint of $\overline{AB}$ , must be on $\overrightarrow{AB}$ , as in the top figure, so $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are one and the same.

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Example Question #2 : Dsq: Understanding Rays

, , and are distinct points.

True or false: $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray.

Statement 1: AB+BC > AC

Statement 2: AB+ AC > BC .

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 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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

We show that both statements together provide insufficient information by giving two scenarios in which both statements are true.

Case 1: , , and are noncollinear. The three points are vertices of a triangle, and by the Triangle Inequality Theorem,

AB+BC > AC and

AB + AC > BC .

Also, since the three points are not on a single line, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are parts of different lines and cannot be the same ray.

Case 2: $\overline{AB}$ with length 2 and midpoint .

Rays

AB+BC =2+1 = 3 and AC = 1 , so AB+BC > AC ; similarly, AB + AC > BC . Also, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray, since they have the same endpoint and is on $\overrightarrow{AB}$ .

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Example Question #3 : Dsq: Understanding Rays

, , and are distinct points.

True or false: $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray.

Statement 1: , , and are collinear.

Statement 2: AB + BC = AC .

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.

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.

Correct answer:

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 prove the rays to be the same or different, as seen in these diagrams:

Rays

In both figures, , , and are collinear, satisfying the condition of Statement 1. But In the top figure, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray, since is on $\overrightarrow{AB}$ ; in the bottom figure, since is not on $\overrightarrow{AB}$ , $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are distinct rays.

Assume Statement 2 alone. Suppose $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are not the same ray. Then one of two things happens:

Case 1: , , and are noncollinear. The three points are vertices of a triangle, and by the triangle inequality,

AB + BC > AC ,

contradicting Statement 2.

Case 2: , , and are collinear. must be between and , as in the bottom figure, since if it were not, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ would be the same ray. By segment addition,

AB + AC = BC

AB + AC+AB = BC+AB

$AB + BC = AC + 2 \cdot AB > AC$ ,

contradicting Statement 2.

By contradiction, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray.

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Example Question #11 : Geometry

What is the measure of $\angle 1$ ?

Statement 1: $\angle 1$ is complementary to an angle that measures $48 ^{\circ }$ .

Statement 2: $\angle 1$ is adjacent to an angle that measures $42 ^{\circ }$ .

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

Complementary angles have degree measures that total $90 ^{\circ }$ , so the measure of an angle complementary to a $48 ^{\circ }$ angle would have measure $(90-48)^{\circ } = 42 ^{\circ }$ . If Statement 1 is assumed, then $m \angle 1= 42 ^{\circ }$ .

Statement 2 gives no useful information. Adjacent angles do not have any numerical relationship; they simply share a ray and a vertex.

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Example Question #2241 : Gmat Quantitative Reasoning

Lines

Note: Figure NOT drawn to scale.

Refer to the above diagram.

What is the measure of $\angle FCB$ ?

Statement 1: $m \angle FBC = 43 ^{\circ }$

Statement 2: $m \angle DCG = 43 ^{\circ }$

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 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.

Correct answer:

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

Explanation:

If we only know that $m \angle FBC = 43 ^{\circ }$ , then we cannot surmise anything from the diagram about the measure of $\angle FCB$ . But $\angle FCB$ and $\angle DCG$ are vertical angles, which must be congruent, so if we know $m \angle DCG = 43 ^{\circ }$ , then $\angle FCB= 43 ^{\circ }$ also.

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Example Question #13 : Geometry

$\angle 1$ and $\angle2$ are supplementary angles. Which one has the greater measure?

Statement 1: $m \angle 1 < 100$

Statement 2: $\angle2$ is an obtuse angle.

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.

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:

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

Explanation:

By definition, if $\angle 1$ and $\angle2$ are supplementary angles, then $m\angle 1 + m\angle 2 = 180$ .

If Statement 1 is assumed and $m \angle 1 < 100$ , then $m\angle 2 > 80$ . This does not answer our question, since, for example, it is possible that $m \angle 1 = 91$ and $m \angle 2 = 89$ , or vice versa.

If Statement 2 is assumed, then $m\angle 2 > 90$ , and subsequently, $m \angle 1 < 90$ ; by transitivity, $m\angle 2 > m\angle 1$ .

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Example Question #2242 : Gmat Quantitative Reasoning

Lines_3

Note: Figure NOT drawn to scale.

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

Statement 1: $m \angle 1 = 30 ^{\circ } + 2 \cdot m \angle 2$

Statement 2: $\angle 3$ is a $130 ^{\circ }$ angle.

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.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. Since $\angle 1$ and $\angle 2$ form a linear pair, their measures total $180^{\circ }$ . Therefore, this fact, along with Statement 1, form a system of linear equations, which can be solved as follows:

$m \angle 1 = 30 ^{\circ } + 2 \cdot m \angle 2$

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

The second equation can be rewritten as

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

and a substitution can be made:

$m \angle 1 = 30 ^{\circ } + 2 \cdot ( 180^{\circ } - m \angle 1)$

$m \angle 1 = 30 ^{\circ } + 360^{\circ } - 2 \cdot m \angle 1$

$m \angle 1 = 390^{\circ } - 2 \cdot m \angle 1$

$m \angle 1 + 2 \cdot m \angle 1 = 390^{\circ } - 2 \cdot m \angle 1 + 2 \cdot m \angle 1$

$3 \cdot m \angle 1 = 390^{\circ }$

$3 \cdot m \angle 1 \div 3 = 390^{\circ } \div 3$

$m \angle 1 = 130^{\circ }$

Assume Statement 2 alone. $\angle 1$ and $\angle 3$ are a pair of vertical angles, which have the same measure, so $m \angle 1=m \angle 3 = 130^{\circ }$ .

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Example Question #3 : Dsq: Calculating An Angle Of A Line

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: $m \angle 1 + m \angle 6 = 181^{\circ }$

Statement 2: $m \angle 7 + m \angle 12 = 179^{\circ }$

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.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. $\angle ADC$ and $\angle 1$ form a linear pair of angles, so their measures total $180^{\circ }$ ; the same holds for $\angle DCB$ and $\angle 6$ . Therefore,

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

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

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

$( m \angle ADC+ m \angle DBC) + (m \angle 1 +m \angle 6 )=360^{\circ }$

$( m \angle ADC+ m \angle DBC) +181^{\circ }=360^{\circ }$

$m \angle ADC+ m \angle DBC=179^{\circ } < 180^{\circ }$

Assume Statement 2 alone. $\angle 12$ and $\angle DAB$ form a linear pair of angles, so their measures total $180^{\circ }$ ; the same holds for $\angle 7$ and $\angle ABC$ . Therefore,

$m \angle 12 + m \angle DAB= 180^{\circ }$

$m \angle7 + m \angle ABC= 180^{\circ }$

$m \angle 12 +m \angle DAB + m \angle 7 + m \angle ABC= 180^{\circ } + 180^{\circ }$

$( m \angle DAB+ m \angle ABC) + (m \angle 7 +m \angle 12 )=360^{\circ }$

$( m \angle DAB+ m \angle ABC) + 179^{\circ } =360^{\circ }$

$m \angle DAB+ m \angle ABC =181^{\circ }$

$\angle DAB$ , $\angle ABC$ , $\angle DCB$ , and $\angle ADC$ are the four angles of Quadrilateral ABCD , so their degree measures total 360. Therefore,

$m \angle ADC+ m \angle DCB + \left (m \angle DAB+ m \angle ABC \right ) =360^{\circ }$

$m \angle ADC+ m \angle DCB + 181 ^{\circ } =360^{\circ }$

$m \angle ADC+ m \angle DCB =179^{\circ }< 180^{\circ }$

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Example Question #6 : Dsq: Calculating An Angle Of A Line

Find the angle made by f(x) and the -axis.

I) f(x) goes through the origin and the point (4,4) .

II) f(x) makes a degree angle between itself and the -axis.

Possible Answers:

Both statements are needed to answer the question.

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

Either statement is sufficient to answer the question.

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

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

Correct answer:

Either statement is sufficient to answer the question.

Explanation:

To find the angle of the line, recall that each quadrant has 90 degrees

I) Tells us that the line has a slope of one. This means that if we make a triangle using our line, the x-axis and a line coming up from the x-axis at 90 degrees we will have a 45/45/90 triangle. Therefore, I) tells us that our angle is 45 degrees.

II) Tells us that the line makes a 45 degree angle between itself and the y-axis. Therefore:

$\theta=90^{\circ}-45^{\circ}=45^{\circ}$

Therfore, we could use either statement.

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Example Question #3 : Dsq: Calculating An Angle Of A Line

Lines_3

Note: Figure NOT drawn to scale.

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

Statement 1: $m \angle 3+ m \angle 6 = 132^{\circ }$

Statement 2: $m \angle 1+ m \angle 4 = 140^{\circ }$

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:

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

Explanation:

Assume Statement 1 alone. $\angle 3$ and $\angle 6$ are a pair of vertical angles and are therefore congruent, so the statement

$m \angle 3+ m \angle 6 = 132^{\circ }$

can be rewritten as

$m \angle 3+ m \angle3 = 132^{\circ }$

$2 \cdot m \angle3 = 132^{\circ }$

$m \angle3 = 66^{\circ }$

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

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

$m \angle 1+m \angle2+ 66 ^{\circ } = 180^{\circ }$

$m \angle 1+m \angle2 = 114^{\circ }$

But without further information, the measure of $\angle 1$ cannot be calculated.

Assume Statement 2 alone. $\angle 1$ and $\angle 4$ are a pair of vertical angles and are therefore congruent, so the statement

$m \angle 1+ m \angle 4 = 140^{\circ }$

can be rewritten as

$m \angle 1+ m \angle 1 = 140^{\circ }$

$2 \cdot m \angle 1 = 140^{\circ }$

$m \angle 1 = 70^{\circ }$

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Dsq: Understanding Rays

Example Question #2 : Dsq: Understanding Rays

Example Question #3 : Dsq: Understanding Rays

Example Question #11 : Geometry

Example Question #2241 : Gmat Quantitative Reasoning

Example Question #13 : Geometry

Example Question #2242 : Gmat Quantitative Reasoning

Example Question #3 : Dsq: Calculating An Angle Of A Line

Example Question #6 : Dsq: Calculating An Angle Of A Line

Example Question #3 : Dsq: Calculating An Angle Of A Line

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