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Example Questions
Example Question #135 : Data Sufficiency Questions
Note: Figure NOT drawn to scale.
Refer to the above figure. Evaluate .
Statement 1:
Statement 2:
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.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
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
Note: Figure NOT drawn to scale.
Refer to the above diagram. What is the measure of ?
Statement 1: is an equilateral triangle.
Statement 2:
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.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
, , 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
Note: Figure NOT drawn to scale.
Refer to the above figure. Evaluate .
Statement 1:
Statement 2:
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.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
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
Note: Figure NOT drawn to scale.
Refer to the above figure. Give the measure of .
Statement 1:
Statement 2:
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.
BOTH statements TOGETHER are insufficient to answer the question.
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
Note: Figure NOT drawn to scale.
Refer to the above diagram. Evaluate .
Statement 1:
Statement 2:
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.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
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
Note: Figure NOT drawn to scale.
Refer to the above diagram. Evaluate .
Statement 1:
Statement 2: is an equilateral triangle.
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.
EITHER statement ALONE is sufficient to answer the question.
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 ,