GMAT Math : DSQ: Calculating an angle of a line

Study concepts, example questions & explanations for GMAT Math

Create An Account

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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:

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.

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.

Report an Error

Example Question #12 : Geometry

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:

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.

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:

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.

Report an Error

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:

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.

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, $m\angle 2 > m\angle 1$ .

Report an Error

Example Question #14 : Geometry

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:

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.

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:

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 $m \angle 1=m \angle 3 = 130^{\circ }$ .

Report an Error

Example Question #15 : Geometry

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.

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

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

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

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

Report an Error

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.

Report an Error

Example Question #16 : Geometry

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:

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.

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:

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

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

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

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

Report an Error

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.

Report an Error

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 $180^{\circ }$ .

Report an Error

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Physics Tutors in Houston, Math Tutors in San Diego, GRE Tutors in San Diego, Computer Science Tutors in New York City, Physics Tutors in Seattle, Calculus Tutors in Washington DC, Spanish Tutors in Philadelphia, ISEE Tutors in Phoenix, MCAT Tutors in Atlanta, English Tutors in Washington DC

Popular Courses & Classes

SAT Courses & Classes in Miami, MCAT Courses & Classes in Washington DC, SSAT Courses & Classes in Philadelphia, LSAT Courses & Classes in Seattle, LSAT Courses & Classes in Houston, Spanish Courses & Classes in Seattle, SSAT Courses & Classes in Boston, Spanish Courses & Classes in Los Angeles, Spanish Courses & Classes in Washington DC, Spanish Courses & Classes in Denver

Popular Test Prep

LSAT Test Prep in Los Angeles, SAT Test Prep in New York City, MCAT Test Prep in Atlanta, ISEE Test Prep in San Diego, ISEE Test Prep in Chicago, SAT Test Prep in Seattle, MCAT Test Prep in San Diego, LSAT Test Prep in San Diego, ACT Test Prep in Philadelphia, ACT Test Prep in Denver

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

Email address:
Your name:
Feedback:

GMAT Math : DSQ: Calculating an angle of a line

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #11 : Geometry

Example Question #12 : Geometry

Example Question #13 : Geometry

Example Question #14 : Geometry

Example Question #15 : Geometry

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

Example Question #16 : Geometry

Example Question #2247 : Gmat Quantitative Reasoning

Example Question #22 : Lines

Example Question #23 : Lines

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details