Card 0 of 5403
Note: Figure NOT drawn to scale.
Refer to the above diagram. Evaluate .
Statement 1:
Statement 2: is an equilateral triangle.
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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Note: Figure NOT drawn to scale.
Refer to the above diagram. What is the measure of ?
Statement 1: is an equilateral triangle.
Statement 2:
,
, 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
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Note: Figure NOT drawn to scale.
Refer to the above diagram. Evaluate .
Statement 1:
Statement 2:
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:
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Note: Figure NOT drawn to scale.
Refer to the above diagram. What is the measure of ?
Statement 1:
Statement 2:
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
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Note: Figure NOT drawn to scale.
Refer to the above diagram. What is the measure of ?
Statement 1: is a
angle.
Statement 2:
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,
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Note: Figure NOT drawn to scale.
Refer to the above diagram. What is the measure of ?
Statement 1:
Statement 2: is a
angle.
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
.
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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:
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,
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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.
Assume Statement 1 alone. Since exists and includes
,
and
are one and the sameāand 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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Note: Figure NOT drawn to scale.
Refer to the above figure. Evaluate .
Statement 1: and
are complementary.
Statement 2:
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.
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Note: Figure NOT drawn to scale.
Refer to the above figure. Evaluate .
Statement 1:
Statement 2:
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.
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Note: Figure NOT drawn to scale.
Refer to the above figure. Evaluate .
Statement 1:
Statement 2:
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
.
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Note: Figure NOT drawn to scale.
Refer to the above figure. Give the measure of .
Statement 1:
Statement 2:
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.
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The equations of two lines are:
Are these lines perpendicular?
Statement 1:
Statement 2:
The lines of the two equations must have slopes that are the opposites of each others reciprocals.
Write each equation in slope-intercept form:
As can be seen, knowing the value of is necessary and sufficient to answer the question. The value of
is irrelevant.
The answer is that Statement 1 alone is sufficient to answer the question, but Statement 2 alone is not sufficient to answer the question.
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Data Sufficiency Question
Is Line A perpendicular to the following line?
Statement 1: The slope of Line A is 3.
Statement 2: Line A passes through the point (2,3).
To determine if two lines are perpendicular, only the slope needs to be considered. The slopes of perpendicular lines are the negative reciprocals of each other. Knowing a single point on the line is not sufficient, as an infinite number of lines can pass through and individual point.
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Refer to the above figure.
True or false:
Statement 1:
Statement 2:
Statement 1 alone establishes by definition that , but does not establish any relationship between
and
.
By Statement 2 alone, since alternating interior angles are congruent, , but no conclusion can be drawn about the relationship of
, since the actual measures of the angles are not given.
Assume both statements are true. By Statement 2, .
and
are corresponding angles formed by a transversal across parallel lines, so
.
is not a right angle, so
.
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Refer to the above figure. True or false:
Statement 1: is equilateral.
Statement 2: Line bisects
.
Statement 1 alone establishes nothing about the angle makes with
, as it is not part of the triangle. Statement 2 alone only establishes that
.
Assume both statements are true. Then is an altitude of an equilateral triangle, making it - and
- perpendicular with the base
- and
.
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Refer to the above figure. True or false:
Statement 1:
Statement 2: Line bisects
.
Assume Statement 1 alone. Then, as a consequence of congruence, and
are congruent. They form a linear pair of angles, so they are also supplementary. Two angles that are both congruent and supplementary must be right angles, so
.
Assume Statement 2 alone. Then , but without any other information about the angles that
or
make with
, it cannot be determined whether
or not.
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Statement 1:
Refer to the above figure. Are the lines perpendicular?
Statement 1:
Statement 2:
Assume Statement 1 alone. The measure of one of the angles formed is
degrees.
Assume Statement 2 alone.
By substituting for
, one angle measure becomes
The marked angles are a linear pair and thus their angle measures add up to 180 degrees; therefore, we can set up an equation:
or
yields illegal angle measures - for example,
yields angle measures
for both angles; the angles are right and the lines are perpendicular.
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Refer to the above figure. Jane chose one of the line segments shown in the above diagram but she will not reveal which one. Which one did she choose?
Statement 1: One of the endpoints of the line segment is .
Statement 2: The line segment includes .
If we know both statements, then we know that the segment can be either or
, since each has endpoint
and each includes
; we can not eliminate either, however.
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How many times does and
intersect?
I) is a linear equation with a slope of
.
II) is quadratic equation with a vertex at
.
When we have a linear equation and a quadratic equation there are only so many times they can intersect. They can intersect 0 times, once, or twice.
I) Gives us the slope of one equation.
II) Gives us the vertex of our quadratic equation.
If you draw a picture, it should be apparent that we don't have enough information to know exactly how many times they intersect. Our quadratic could be facing up or down, and our linear equation could go straight through both arms, or it could miss it entirely. Therfore, neither statement is sufficient.
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