GMAT Math : DSQ: Calculating an angle of a line

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #135 : Data Sufficiency Questions

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above figure. Evaluate .

Statement 1: 

Statement 2: 

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.  and  are congruent legs of right triangle , so their acute angles, one of which is , measure  and  form a pair of vertical, and consequently, congruent, angles, so .

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

Example Question #25 : Lines

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above diagram. What is the measure of  ?

Statement 1:  is an equilateral triangle.

Statement 2: 

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:

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

Assume Statement 1 alone. The angles of an equilateral triangle all measure , so  and  form a pair of vertical angles, so they are congruent, and consequently, . Therefore,

But with no further information,  cannot be calculated.

Assume Statement 2 alone. It follows that 

Again, with no further information,  cannot be calculated.

Assume both statements to be true.  as a result of Statement 1, and  from Statement 2, so

Example Question #141 : Data Sufficiency Questions

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above figure. Evaluate .

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.

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

Without further information, no other angle measures, including that of , can be found.

Assume Statement 2 alone. , and  together form a straight angle, so their degree measures total 

Without further information, no other angle measures, including that of , can be found.

However, if both statements are assumed to be true, it follows from Statements 2 and 1 respectively, as seen before, that  and , so

.

Example Question #27 : Lines

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above figure. Give the measure of .

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. 

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: 

 and  are vertical from  and , respectively, so  and , and 

 

Case 2: 

 

The conditions of both statements are met, but  assumes a different value in each scenario.

Example Question #28 : Lines

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

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.

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

However, without any further information, we cannot determine the sum of the measures of  and .

Assume Statement 2 alone.  , and  together form a straight angle, so their measures total ; therefore,

Again, without any further information, we cannot determine the sum of the measures of  and .

Assume both statements are true. Since the measures of  and  can be calculated from Statements 1 and 2, respectively. We can add them:

Example Question #29 : Lines

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

Statement 1: 

Statement 2:  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.  and  are a pair of vertical angles, as are  and . Therefore, 

 

By substitution,

.

Assume Statement 2 alone. The angles of an equilateral triangle all measure , so .

,  , and  together form a straight angle, so , 

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