GMAT Math : Data-Sufficiency Questions

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Example Questions

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

$\angle 1$ $\displaystyle \angle 1$ and $\angle2$ $\displaystyle \angle2$ are supplementary angles. Which one has the greater measure?

Statement 1: $m \angle 1 < 100$ $\displaystyle m \angle 1 < 100$

Statement 2: $\angle2$ $\displaystyle \angle2$ 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.

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.

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 $\angle 1$ $\displaystyle \angle 1$ and $\angle2$ $\displaystyle \angle2$ are supplementary angles, then $m\angle 1 + m\angle 2 = 180$ $\displaystyle m\angle 1 + m\angle 2 = 180$.

If Statement 1 is assumed and $m \angle 1 < 100$ $\displaystyle m \angle 1 < 100$, then $m\angle 2 > 80$ $\displaystyle m\angle 2 > 80$. This does not answer our question, since, for example, it is possible that $m \angle 1 = 91$ $\displaystyle m \angle 1 = 91$ and $m \angle 2 = 89$ $\displaystyle m \angle 2 = 89$, or vice versa.

If Statement 2 is assumed, then $m\angle 2 > 90$ $\displaystyle m\angle 2 > 90$, and subsequently, $m \angle 1 < 90$ $\displaystyle m \angle 1 < 90$; by transitivity, $m\angle 2 > m\angle 1$ $\displaystyle m\angle 2 > m\angle 1$.

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Example Question #131 : Data Sufficiency Questions

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above diagram. What is the measure of $\angle 1$ $\displaystyle \angle 1$?

Statement 1: $m \angle 1 = 30 ^{\circ } + 2 \cdot m \angle 2$ $\displaystyle m \angle 1 = 30 ^{\circ } + 2 \cdot m \angle 2$

Statement 2: $\angle 3$ $\displaystyle \angle 3$ is a $130 ^{\circ }$ $\displaystyle 130 ^{\circ }$ angle.

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 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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. Since $\angle 1$ $\displaystyle \angle 1$ and $\angle 2$ $\displaystyle \angle 2$ form a linear pair, their measures total $180^{\circ }$ $\displaystyle 180^{\circ }$. Therefore, this fact, along with Statement 1, form a system of linear equations, which can be solved as follows:

$m \angle 1 = 30 ^{\circ } + 2 \cdot m \angle 2$ $\displaystyle m \angle 1 = 30 ^{\circ } + 2 \cdot m \angle 2$

$m \angle 1+ m \angle 2 = 180^{\circ }$ $\displaystyle m \angle 1+ m \angle 2 = 180^{\circ }$

The second equation can be rewritten as

$m \angle 2= 180^{\circ } - m \angle 1$ $\displaystyle m \angle 2= 180^{\circ } - m \angle 1$

and a substitution can be made:

$m \angle 1 = 30 ^{\circ } + 2 \cdot ( 180^{\circ } - m \angle 1)$ $\displaystyle m \angle 1 = 30 ^{\circ } + 2 \cdot ( 180^{\circ } - m \angle 1)$

$m \angle 1 = 30 ^{\circ } + 360^{\circ } - 2 \cdot m \angle 1$ $\displaystyle m \angle 1 = 30 ^{\circ } + 360^{\circ } - 2 \cdot m \angle 1$

$m \angle 1 = 390^{\circ } - 2 \cdot m \angle 1$ $\displaystyle m \angle 1 = 390^{\circ } - 2 \cdot m \angle 1$

$m \angle 1 + 2 \cdot m \angle 1 = 390^{\circ } - 2 \cdot m \angle 1 + 2 \cdot m \angle 1$ $\displaystyle m \angle 1 + 2 \cdot m \angle 1 = 390^{\circ } - 2 \cdot m \angle 1 + 2 \cdot m \angle 1$

$3 \cdot m \angle 1 = 390^{\circ }$ $\displaystyle 3 \cdot m \angle 1 = 390^{\circ }$

$3 \cdot m \angle 1 \div 3 = 390^{\circ } \div 3$ $\displaystyle 3 \cdot m \angle 1 \div 3 = 390^{\circ } \div 3$

$m \angle 1 = 130^{\circ }$ $\displaystyle m \angle 1 = 130^{\circ }$

Assume Statement 2 alone. $\angle 1$ $\displaystyle \angle 1$ and $\angle 3$ $\displaystyle \angle 3$ are a pair of vertical angles, which have the same measure, so $m \angle 1=m \angle 3 = 130^{\circ }$ $\displaystyle m \angle 1=m \angle 3 = 130^{\circ }$.

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Example Question #132 : Data Sufficiency Questions

Lines_4

Note: You may assume that $\displaystyle t$ and $\displaystyle u$ are not parallel lines, but you may not assume that $\displaystyle l$ and $\displaystyle m$ are parallel lines unless it is specifically stated.

Refer to the above diagram. Is the sum of the measures of $\angle ADC$ $\displaystyle \angle ADC$ and $\angle DCB$ $\displaystyle \angle DCB$ less than, equal to, or greater than $180^{\circ }$ $\displaystyle 180^{\circ }$?

Statement 1: $m \angle 1 + m \angle 6 = 181^{\circ }$ $\displaystyle m \angle 1 + m \angle 6 = 181^{\circ }$

Statement 2: $m \angle 7 + m \angle 12 = 179^{\circ }$ $\displaystyle m \angle 7 + m \angle 12 = 179^{\circ }$

Possible Answers:

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.

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. $\angle ADC$ $\displaystyle \angle ADC$ and $\angle 1$ $\displaystyle \angle 1$ form a linear pair of angles, so their measures total $180^{\circ }$ $\displaystyle 180^{\circ }$; the same holds for $\angle DCB$ $\displaystyle \angle DCB$ and $\angle 6$ $\displaystyle \angle 6$. Therefore,

$m \angle 1 + m \angle ADC = 180^{\circ }$ $\displaystyle m \angle 1 + m \angle ADC = 180^{\circ }$

$m \angle 6 + m \angle DBC = 180^{\circ }$ $\displaystyle m \angle 6 + m \angle DBC = 180^{\circ }$

$m \angle 1 +m \angle ADC+ m \angle 6 + m \angle DBC = 180^{\circ } + 180^{\circ }$ $\displaystyle m \angle 1 +m \angle ADC+ m \angle 6 + m \angle DBC = 180^{\circ } + 180^{\circ }$

$( m \angle ADC+ m \angle DBC) + (m \angle 1 +m \angle 6 )=360^{\circ }$ $\displaystyle ( m \angle ADC+ m \angle DBC) + (m \angle 1 +m \angle 6 )=360^{\circ }$

$( m \angle ADC+ m \angle DBC) +181^{\circ }=360^{\circ }$ $\displaystyle ( m \angle ADC+ m \angle DBC) +181^{\circ }=360^{\circ }$

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

Assume Statement 2 alone. $\angle 12$ $\displaystyle \angle 12$ and $\angle DAB$ $\displaystyle \angle DAB$ form a linear pair of angles, so their measures total $180^{\circ }$ $\displaystyle 180^{\circ }$; the same holds for $\angle 7$ $\displaystyle \angle 7$ and $\angle ABC$ $\displaystyle \angle ABC$. Therefore,

$m \angle 12 + m \angle DAB= 180^{\circ }$ $\displaystyle m \angle 12 + m \angle DAB= 180^{\circ }$

$m \angle7 + m \angle ABC= 180^{\circ }$ $\displaystyle m \angle7 + m \angle ABC= 180^{\circ }$

$m \angle 12 +m \angle DAB + m \angle 7 + m \angle ABC= 180^{\circ } + 180^{\circ }$ $\displaystyle m \angle 12 +m \angle DAB + m \angle 7 + m \angle ABC= 180^{\circ } + 180^{\circ }$

$( m \angle DAB+ m \angle ABC) + (m \angle 7 +m \angle 12 )=360^{\circ }$ $\displaystyle ( m \angle DAB+ m \angle ABC) + (m \angle 7 +m \angle 12 )=360^{\circ }$

$( m \angle DAB+ m \angle ABC) + 179^{\circ } =360^{\circ }$ $\displaystyle ( m \angle DAB+ m \angle ABC) + 179^{\circ } =360^{\circ }$

$m \angle DAB+ m \angle ABC =181^{\circ }$ $\displaystyle m \angle DAB+ m \angle ABC =181^{\circ }$

$\angle DAB$ $\displaystyle \angle DAB$, $\angle ABC$ $\displaystyle \angle ABC$, $\angle DCB$ $\displaystyle \angle DCB$, and $\angle ADC$ $\displaystyle \angle ADC$ are the four angles of Quadrilateral ABCD $\displaystyle ABCD$, so their degree measures total 360. Therefore,

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

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

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

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

Find the angle made by f(x) $\displaystyle f(x)$ and the $\displaystyle x$-axis.

I) f(x) $\displaystyle f(x)$ goes through the origin and the point (4,4) $\displaystyle (4,4)$.

II) f(x) $\displaystyle f(x)$ makes a $\displaystyle 45$ degree angle between itself and the $\displaystyle y$-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:

$\theta=90^{\circ}-45^{\circ}=45^{\circ}$ $\displaystyle \theta=90^{\circ}-45^{\circ}=45^{\circ}$

Therfore, we could use either statement.

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Example Question #3 : 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 $\angle 1$ $\displaystyle \angle 1$ ?

Statement 1: $m \angle 3+ m \angle 6 = 132^{\circ }$ $\displaystyle m \angle 3+ m \angle 6 = 132^{\circ }$

Statement 2: $m \angle 1+ m \angle 4 = 140^{\circ }$ $\displaystyle m \angle 1+ m \angle 4 = 140^{\circ }$

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.

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.

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. $\angle 3$ $\displaystyle \angle 3$ and $\angle 6$ $\displaystyle \angle 6$ are a pair of vertical angles and are therefore congruent, so the statement

$m \angle 3+ m \angle 6 = 132^{\circ }$ $\displaystyle m \angle 3+ m \angle 6 = 132^{\circ }$

can be rewritten as

$m \angle 3+ m \angle3 = 132^{\circ }$ $\displaystyle m \angle 3+ m \angle3 = 132^{\circ }$

$2 \cdot m \angle3 = 132^{\circ }$ $\displaystyle 2 \cdot m \angle3 = 132^{\circ }$

$m \angle3 = 66^{\circ }$ $\displaystyle m \angle3 = 66^{\circ }$

$\angle 1$ $\displaystyle \angle 1$, $\angle 2$ $\displaystyle \angle 2$, and $\angle 3$ $\displaystyle \angle 3$ together form a straight angle, so their measures total $180^{\circ }$ $\displaystyle 180^{\circ }$; therefore,

$m \angle 1+m \angle2+ m \angle 3 = 180^{\circ }$ $\displaystyle m \angle 1+m \angle2+ m \angle 3 = 180^{\circ }$

$m \angle 1+m \angle2+ 66 ^{\circ } = 180^{\circ }$ $\displaystyle m \angle 1+m \angle2+ 66 ^{\circ } = 180^{\circ }$

$m \angle 1+m \angle2 = 114^{\circ }$ $\displaystyle m \angle 1+m \angle2 = 114^{\circ }$

But without further information, the measure of $\angle 1$ $\displaystyle \angle 1$ cannot be calculated.

Assume Statement 2 alone. $\angle 1$ $\displaystyle \angle 1$ and $\angle 4$ $\displaystyle \angle 4$ are a pair of vertical angles and are therefore congruent, so the statement

$m \angle 1+ m \angle 4 = 140^{\circ }$ $\displaystyle m \angle 1+ m \angle 4 = 140^{\circ }$

can be rewritten as

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

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

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

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Example Question #2242 : Gmat Quantitative Reasoning

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above figure. Evaluate $m \angle 1 + m \angle 2$ $\displaystyle m \angle 1 + m \angle 2$.

Statement 1: $\angle 4$ $\displaystyle \angle 4$ and $\angle 5$ $\displaystyle \angle 5$ are complementary.

Statement 2: $OD \cdot \sqrt{2} = OE$ $\displaystyle OD \cdot \sqrt{2} = OE$

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.

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

Explanation:

Assume Statement 1 alone. $\angle 5$ $\displaystyle \angle 5$ and $\angle 6$ $\displaystyle \angle 6$ are vertical from $\angle 1$ $\displaystyle \angle 1$ and $\angle 2$ $\displaystyle \angle 2$, respectively, so $m \angle 1 = m \angle 2$ $\displaystyle m \angle 1 = m \angle 2$ and $m \angle 2 = m \angle 6$ $\displaystyle m \angle 2 = m \angle 6$. $\angle 5$ $\displaystyle \angle 5$ and $\angle 6$ $\displaystyle \angle 6$ form a complementary pair, so, by definition

$m \angle 5 + m \angle 6 = 90^{\circ}$ $\displaystyle m \angle 5 + m \angle 6 = 90^{\circ}$

and by substitution,

$m \angle 1 + m \angle 2 = 90^{\circ}$ $\displaystyle m \angle 1 + m \angle 2 = 90^{\circ}$.

Assume Statement 2 alone. Since $\bigtriangleup ODE$ $\displaystyle \bigtriangleup ODE$ is a right triangle whose hypotenuse is $\sqrt{2}$ $\displaystyle \sqrt{2}$ times as long as a leg, it follows that $\bigtriangleup ODE$ $\displaystyle \bigtriangleup ODE$ is a 45-45-90 triangle, so $m \angle DOE = 45^{\circ }$ $\displaystyle m \angle DOE = 45^{\circ }$.

$\angle 1$ $\displaystyle \angle 1$, $\angle 2$ $\displaystyle \angle 2$, $\angle 3$ $\displaystyle \angle 3$, and $\angle DOE$ $\displaystyle \angle DOE$ together form a straight angle, so their degree measures total $180^{\circ }$ $\displaystyle 180^{\circ }$.

$m \angle 1 + m \angle 2+ m \angle 3+m \angle DOE = 180^{\circ }$ $\displaystyle m \angle 1 + m \angle 2+ m \angle 3+m \angle DOE = 180^{\circ }$

$m \angle 1 + m \angle 2+ m \angle 3+ 45^{\circ } = 180^{\circ }$ $\displaystyle m \angle 1 + m \angle 2+ m \angle 3+ 45^{\circ } = 180^{\circ }$

$m \angle 1 + m \angle 2+ m \angle 3 = 135^{\circ }$ $\displaystyle m \angle 1 + m \angle 2+ m \angle 3 = 135^{\circ }$

But without further information, the sum of the degree measures of only $\angle 1$ $\displaystyle \angle 1$ and $\angle 2$ $\displaystyle \angle 2$ cannot be calculated.

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Example Question #22 : Lines

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above diagram. What is the measure of $\angle 1$ $\displaystyle \angle 1$ ?

Statement 1: $\angle 6$ $\displaystyle \angle 6$ is a $54 ^{\circ }$ $\displaystyle 54 ^{\circ }$ angle.

Statement 2: $\angle 2 \cong \angle 3$ $\displaystyle \angle 2 \cong \angle 3$

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 $\angle 1$ $\displaystyle \angle 1$.

$\angle 6$ $\displaystyle \angle 6$, $\angle 1$ $\displaystyle \angle 1$, and $\angle 2$ $\displaystyle \angle 2$ together form a $180^{\circ }$ $\displaystyle 180^{\circ }$angle; therefore,

$m \angle 1+m \angle2+ m \angle 6 = 180^{\circ }$ $\displaystyle m \angle 1+m \angle2+ m \angle 6 = 180^{\circ }$

$m \angle 6 = 54^{\circ }$ $\displaystyle m \angle 6 = 54^{\circ }$, so by substitution,

$m \angle 1+m \angle2+ 54^{\circ } = 180^{\circ }$ $\displaystyle m \angle 1+m \angle2+ 54^{\circ } = 180^{\circ }$

$m \angle 1+m \angle2= 126^{\circ }$ $\displaystyle m \angle 1+m \angle2= 126^{\circ }$

But with no further information, the measure of $\angle 1$ $\displaystyle \angle 1$ cannot be calculated.

Statement 2 alone gives insufficient information for a similar reason. $\angle 1$ $\displaystyle \angle 1$, $\angle 2$ $\displaystyle \angle 2$, and $\angle 3$ $\displaystyle \angle 3$ together form a $180^{\circ }$ $\displaystyle 180^{\circ }$ angle; therefore,

$m \angle 1+m \angle2+ m \angle 3 = 180^{\circ }$ $\displaystyle m \angle 1+m \angle2+ m \angle 3 = 180^{\circ }$

Since $\angle 2 \cong \angle 3$ $\displaystyle \angle 2 \cong \angle 3$, we can rewrite this statement as

$m \angle 1+m \angle2+ m \angle 2 = 180^{\circ }$ $\displaystyle m \angle 1+m \angle2+ m \angle 2 = 180^{\circ }$

$m \angle 1+2 \cdot m \angle2 = 180^{\circ }$ $\displaystyle m \angle 1+2 \cdot m \angle2 = 180^{\circ }$

Again, with no further information, the measure of $\angle 1$ $\displaystyle \angle 1$ cannot be calculated.

Assume both statements to be true. $\angle 2$ $\displaystyle \angle 2$ and $\angle 6$ $\displaystyle \angle 6$ are a pair of vertical angles, so $\angle 2 \cong \angle 6$ $\displaystyle \angle 2 \cong \angle 6$, and $m \angle 2 = m \angle 6 = 54^{\circ }$ $\displaystyle m \angle 2 = m \angle 6 = 54^{\circ }$. Since $\angle 2 \cong \angle 3$ $\displaystyle \angle 2 \cong \angle 3$, then $m \angle 3 = m \angle 2 = 54^{\circ }$ $\displaystyle m \angle 3 = m \angle 2 = 54^{\circ }$. Also,

$m \angle 1+m \angle2+ m \angle 3 = 180^{\circ }$ $\displaystyle m \angle 1+m \angle2+ m \angle 3 = 180^{\circ }$

By substitution,

$m \angle 1+ 54^{\circ }+ 54^{\circ } = 180^{\circ }$ $\displaystyle m \angle 1+ 54^{\circ }+ 54^{\circ } = 180^{\circ }$

$m \angle 1+108^{\circ } = 180^{\circ }$ $\displaystyle m \angle 1+108^{\circ } = 180^{\circ }$

$m \angle 1=72^{\circ }$ $\displaystyle m \angle 1=72^{\circ }$

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Example Question #23 : Lines

Lines_4

Statement 1: There exists a point $\displaystyle Z$ such that $\displaystyle A$ lies on $\overline{ZD}$ $\displaystyle \overline{ZD}$ and $\displaystyle B$ lies on $\overline{ZC}$ $\displaystyle \overline{ZC}$.

Statement 2: Quadrilateral ABCD $\displaystyle ABCD$ 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 $\overline{ZD}$ $\displaystyle \overline{ZD}$ exists and includes $\displaystyle A$, $\overleftrightarrow{ZD}$ $\displaystyle \overleftrightarrow{ZD}$ and $\overleftrightarrow{DA}$ $\displaystyle \overleftrightarrow{DA}$ are one and the same—and this is $\displaystyle l$. Similarly, $\overleftrightarrow{ZC}$ $\displaystyle \overleftrightarrow{ZC}$ is $\displaystyle m$. This means that $\displaystyle l$ and $\displaystyle m$ have a point of intersection, which is $\displaystyle Z$. Since $\displaystyle A$ falls between $\displaystyle Z$ and $\displaystyle C$ and $\displaystyle B$ falls between $\displaystyle Z$ and $\displaystyle D$, the lines intersect on the side of $\displaystyle t$ that includes points $\displaystyle A$ and $\displaystyle B$. By Euclid's Fifth Postulate, the sum of the measures of $\angle ADC$ $\displaystyle \angle ADC$ and $\angle DCB$ $\displaystyle \angle DCB$ is less than $180^{\circ }$ $\displaystyle 180^{\circ }$.

Assume Statement 2 alone. Since it is given that $t \nparallel u$ $\displaystyle t \nparallel u$, the other two sides, $\overline{AD}$ $\displaystyle \overline{AD}$ and $\overline{BC}$ $\displaystyle \overline{BC}$ are parallel if and only if Quadrilateral ABCD $\displaystyle ABCD$ is a trapezoid, which it is not. Therefore, $\overline{AD}$ $\displaystyle \overline{AD}$ and $\overline{BC}$ $\displaystyle \overline{BC}$ are not parallel, and the sum of the degree measures of same-side interior angles $\angle ADC$ $\displaystyle \angle ADC$ and $\angle DCB$ $\displaystyle \angle DCB$ is not equal to $180^{\circ }$ $\displaystyle 180^{\circ }$. However, without further information, it is impossible to determine whether the sum of the measures is less than or greater than $180^{\circ }$ $\displaystyle 180^{\circ }$.

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Example Question #135 : Data Sufficiency Questions

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above figure. Evaluate $m \angle 3$ $\displaystyle m \angle 3$.

Statement 1: $\overline{BO} \cong \overline{BC}$ $\displaystyle \overline{BO} \cong \overline{BC}$

Statement 2: $m \angle 5 = 42^{\circ }$ $\displaystyle m \angle 5 = 42^{\circ }$

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. $\overline{BO}$ $\displaystyle \overline{BO}$ and $\overline{BC}$ $\displaystyle \overline{BC}$ are congruent legs of right triangle $\bigtriangleup OBC$ $\displaystyle \bigtriangleup OBC$, so their acute angles, one of which is $\angle BOC$ $\displaystyle \angle BOC$, measure $45^{\circ }$ $\displaystyle 45^{\circ }$. $\angle BOC$ $\displaystyle \angle BOC$ and $\angle 3$ $\displaystyle \angle 3$ form a pair of vertical, and consequently, congruent, angles, so $m \angle 3 = 45^{\circ }$ $\displaystyle m \angle 3 = 45^{\circ }$.

Statement 2 alone gives insufficient information, as $\angle 3$ $\displaystyle \angle 3$ and $\angle 5$ $\displaystyle \angle 5$ has no particular relationship that would lead to an arithmetic relationship between their angle measures.

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Example Question #25 : Lines

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above diagram. What is the measure of $\angle 1$ $\displaystyle \angle 1$ ?

Statement 1: $\bigtriangleup OPQ$ $\displaystyle \bigtriangleup OPQ$ is an equilateral triangle.

Statement 2: $m \angle 3 = 65^{\circ }$ $\displaystyle m \angle 3 = 65^{\circ }$

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:

$m \angle 1+m \angle2+ m \angle 3 = 180^{\circ }$ $\displaystyle m \angle 1+m \angle2+ m \angle 3 = 180^{\circ }$

Assume Statement 1 alone. The angles of an equilateral triangle all measure $60^{\circ }$ $\displaystyle 60^{\circ }$, so $m \angle POQ = 60 ^{\circ }$ $\displaystyle m \angle POQ = 60 ^{\circ }$; $\angle POQ$ $\displaystyle \angle POQ$ and $\angle 2$ $\displaystyle \angle 2$ form a pair of vertical angles, so they are congruent, and consequently, $m \angle 2 = 60^{\circ }$ $\displaystyle m \angle 2 = 60^{\circ }$. Therefore,

$m \angle 1+ 60^{\circ } + m \angle 3 = 180^{\circ }$ $\displaystyle m \angle 1+ 60^{\circ } + m \angle 3 = 180^{\circ }$

$m \angle 1 + m \angle 3 = 120^{\circ }$ $\displaystyle m \angle 1 + m \angle 3 = 120^{\circ }$

But with no further information, $m \angle 1$ $\displaystyle m \angle 1$ cannot be calculated.

Assume Statement 2 alone. It follows that

$m \angle 1+m \angle2+ 65^{\circ }= 180^{\circ }$ $\displaystyle m \angle 1+m \angle2+ 65^{\circ }= 180^{\circ }$

$m \angle 1+m \angle2 =115^{\circ }$ $\displaystyle m \angle 1+m \angle2 =115^{\circ }$

Again, with no further information, $m \angle 1$ $\displaystyle m \angle 1$ cannot be calculated.

Assume both statements to be true. $m \angle 2 = 60^{\circ }$ $\displaystyle m \angle 2 = 60^{\circ }$ as a result of Statement 1, and $m \angle 3 = 65^{\circ }$ $\displaystyle m \angle 3 = 65^{\circ }$ from Statement 2, so

$m \angle 1+60^{\circ } + 65^{\circ }= 180^{\circ }$ $\displaystyle m \angle 1+60^{\circ } + 65^{\circ }= 180^{\circ }$

$m \angle 1+125^{\circ }= 180^{\circ }$ $\displaystyle m \angle 1+125^{\circ }= 180^{\circ }$

$m \angle 1=55^{\circ }$ $\displaystyle m \angle 1=55^{\circ }$

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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.

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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.

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