GMAT Math : DSQ: Calculating an angle of a line

Study concepts, example questions & explanations for GMAT Math

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Example Question #1 : Dsq: Calculating An Angle Of A Line

What is the measure of ?

Statement 1:  is complementary to an angle that measures .

Statement 2:  is adjacent to an angle that measures .

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 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.

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 , so the measure of an angle complementary to a  angle would have measure . If Statement 1 is assumed, then .

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

Example Question #2 : Dsq: Calculating An Angle Of A Line

Lines

Note: Figure NOT drawn to scale.

Refer to the above diagram.

What is the measure of  ?

Statement 1: 

Statement 2: 

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.

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.

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 , then we cannot surmise anything from the diagram about the measure of . But  and  are vertical angles, which must be congruent, so if we know , then  also.

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

 and  are supplementary angles. Which one has the greater measure?

Statement 1:  

Statement 2:  is an obtuse angle.

Possible Answers:

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 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.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 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  and  are supplementary angles, then .

If Statement 1 is assumed and , then . This does not answer our question, since, for example, it is possible that  and , or vice versa.

If Statement 2 is assumed, then , and subsequently, ; by transitivity, .

Example Question #4 : 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 ?

Statement 1: 

Statement 2:  is a  angle.

Possible Answers:

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.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. Since  and  form a linear pair, their measures total . Therefore, this fact, along with Statement 1, form a system of linear equations, which can be solved as follows:

The second equation can be rewritten as 

and a substitution can be made:

Assume Statement 2 alone.  and  are a pair of vertical angles, which have the same measure, so  .

Example Question #5 : 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  and  less than, equal to, or greater than ?

Statement 1: 

Statement 2: 

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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone.  and  form a linear pair of angles, so their measures total ; the same holds for  and . Therefore,

Assume Statement 2 alone.  and  form a linear pair of angles, so their measures total ; the same holds for  and . Therefore,

, , and  are the four angles of Quadrilateral , so their degree measures total 360. Therefore,

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

Find the angle made by  and the -axis.

I)  goes through the origin and the point .

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

Possible Answers:

Both statements are needed to answer the question.

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

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

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

Either statement is 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:

Therfore, we could use either statement.

Example Question #7 : 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  ?

Statement 1: 

Statement 2: 

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:

Assume Statement 1 alone.  and  are a pair of vertical angles and are therefore congruent, so the statement

can be rewritten as

 , and  together form a straight angle, so their measures total ; therefore,

But without further information, the measure of  cannot be calculated.

Assume Statement 2 alone.  and  are a pair of vertical angles and are therefore congruent, so the statement

can be rewritten as

Example Question #8 : Dsq: Calculating An Angle Of A Line

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above figure. Evaluate .

Statement 1:  and  are complementary.

Statement 2: 

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.

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.

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.  and  are vertical from  and , respectively, so  and  and  form a complementary pair, so, by definition 

and by substitution,

.

Assume Statement 2 alone. Since  is a right triangle whose hypotenuse is  times as long as a leg, it follows that  is a 45-45-90 triangle, so .

 , and  together form a straight angle, so their degree measures total 

But without further information, the sum of the degree measures of only  and  cannot be calculated.

Example Question #9 : 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  ?

Statement 1:  is a  angle.

Statement 2: 

Possible Answers:

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 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 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 .

, and  together form a angle; therefore,

, so by substitution,

But with no further information, the measure of  cannot be calculated.

Statement 2 alone gives insufficient information for a similar reason. , , and  together form a  angle; therefore,

Since , we can rewrite this statement as

Again, with no further information, the measure of  cannot be calculated.

Assume both statements to be true.   and  are a pair of vertical angles, so , and . Since , then . Also, 

By substitution,

 

Example Question #10 : 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  and  less than, equal to, or greater than ?

Statement 1: There exists a point  such that  lies on  and  lies on .

Statement 2: Quadrilateral  is not a trapezoid.

Possible Answers:

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.

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 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  exists and includes ,   and  are one and the same—and this is . Similarly,  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  and  is less than .

Assume Statement 2 alone. Since it is given that , the other two sides,  and  are parallel if and only if Quadrilateral  is a trapezoid, which it is not. Therefore,   and  are not parallel, and the sum of the degree measures of same-side interior angles   and  is not equal to . However, without further information, it is impossible to determine whether the sum of the measures is less than or greater than .

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