Lines - GMAT Quantitative Reasoning

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Question

A right triangle is given with a missing value of . It is stated that the triangle is an acute right triangle with angles $50^\circ$ and . What is a possible value of in degrees?

Answer

It is important to recall that all triangles add to 180 degrees and a right triangle contains one angle that is equal to 90 degrees. Therefore, in this particular problem we can write the following equation to solve for the missing variable.

$\90^\circ+50^\circ+2t=180^\circ \140^\circ-2t=180^\circ \2t=180^\circ-140^\circ \2t=40^\circ \t=20$

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Question

What is the measure of an angle complementary to a $72 ^{\circ }$ angle?

Answer

Complementary angles have degree measures that total $90 ^{\circ }$ , so the measure of an angle complementary to a $72 ^{\circ }$ angle would have measure $(90-72)^{\circ } = 18 ^{\circ }$ .

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Question

What is the measure of an angle supplementary to a $66 ^{\circ }$ angle?

Answer

Supplementary angles have degree measures that total $180 ^{\circ }$ , so the measure of an angle complementary to a $66 ^{\circ }$ angle would have measure $\left ( 180- 66 \right )^{\circ } = 114^{\circ }$ .

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Question

What is the measure of an angle congruent to a $48 ^{\circ }$ angle?

Answer

Two angles are congruent if they have the same degree measure, so an angle will be congruent to a $48 ^{\circ }$ angle if its measure is also $48 ^{\circ }$ .

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Question

What is the measure of an angle that is supplementary to a $68^{\circ}$ angle?

Answer

Supplementary angles have degree measures that total $180^{\circ}$ , so an angle supplementary to $68^{\circ}$ would measure $180^{\circ}-68^{\circ}=112^{\circ}$ .

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Question

What is the measure of an angle that is complementary to a $42^{\circ }$ angle?

Answer

Complementary angles have degree measures that total $90^{\circ}$ , so an angle complementary to $42^{\circ}$ would measure $90^{\circ}-42^{\circ}=48^{\circ}$ .

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Question

What is the measure of an angle congruent to a $58^{\circ}$ angle?

Answer

Congruent angles have degree measures that are equal, so an angle congruent to $58^{\circ}$ is $58^{\circ}$ .

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Question

What is the measure of an angle that is supplementary to a $80^{\circ}$ angle?

Answer

Supplementary angles have degree measures that total $180^{\circ }$ . Since we have an $80^{\circ}$ angle, the supplementary angle would measure $180^{\circ}-80^{\circ }=100^{\circ}$

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Question

Which of the following angles is complementary to an $18^{\circ}$ angle?

Answer

Complementary angles have degree measures that total $90^{\circ }$ . Since we have an $18^{\circ}$ angle, the supplementary angle would measure $90^{\circ}-18^{\circ }=72^{\circ}$

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Question

Which of the following angles is congruent to a $119^{\circ}$ angle?

Answer

Congruent angles have the same degree measure, so an angle congruent to a $119^{\circ}$ angle would also measure $119^{\circ}$ .

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Question

What is the measurement of an angle that is congruent to an $89^{\circ }$ angle?

Answer

Two angles are congruent if they have the same degree measure. Therefore, an angle congruent to an $89^{\circ }$ angle also measures $89^{\circ }$ .

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Question

What is the measurement of an angle that is supplementary to a $16^{\circ}$ angle?

Answer

Two angles are supplementary if the total of their degree measures is $180^{\circ }$ . Therefore, an angle supplementary to a $16^{\circ}$ angle measures $180^{\circ }-16^{\circ}=164^{\circ}$ .

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Question

What is the measure of an angle that is complementary to a $70^{\circ}$ angle?

Answer

Two angles are complementary if the total of their degree measures is $90^{\circ }$ . Therefore, an angle complementary to a $70^{\circ}$ angle measures $90^{\circ }-70^{\circ}=20^{\circ}$ .

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Question

Find the equation of the line that is perpendicular to the line connecting the points $\dpi{100} \small (0,-4)\ and\ (-1,-7)$ .

Answer

Lines are perpendicular if their slopes are negative reciprocals of each other. First we need to find the slope of the line in the question stem.

$slope = \frac{rise}{run} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{-7 + 4}{-1 - 0} = \frac{-3}{-1} = 3$

The negative reciprocal of 3 is $\dpi{100} \small -\frac{1}{3}$ , so our answer will have a slope of $\dpi{100} \small -\frac{1}{3}$ . Let's go through the answer choices and see.

$\dpi{100} \small y=3x-1$ : This line is of the form $\dpi{100} \small y=mx+b$ , where $\dpi{100} \small m$ is the slope. The slope is 3, so this line is parallel, not perpendicular, to our line in question.

$\dpi{100} \small y=-4x+8$ : The slope here is $\dpi{100} \small -4$ , also wrong.

$\dpi{100} \small y=\frac{x}{3}+1$ : The slope of this line is $\dpi{100} \small \frac{1}{3}$ . This is the reciprocal, but not the negative reciprocal, so this is also incorrect.

The line between the points $\dpi{100} \small (3,0)\ and\ (-3,2)$ : $\dpi{100} \small slope = \frac{2}{(-3-3)}=\frac{2}{-6}=-\frac{1}{3}$ .

This is the correct answer! Let's check the last answer choice as well.

The line between points $\dpi{100} \small (0,0)\ and\ (2,2)$ : $\dpi{100} \small slope = \frac{2}{2}=1$ , which is incorrect.

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Question

Determine whether the lines with equations and are perpendicular.

Answer

If two equations are perpendicular, then they will have inverse negative slopes of each other. So if we compare the slopes of the two equations, then we can find the answer. For the first equation we have $3y+x=2\Rightarrow y=\frac{-1}{3}x+\frac{2}{3}$

so the slope is $-\frac{1}{3}$ .

So for the equations to be perpendicular, the other equation needs to have a slope of 3. For the second equation, we have

$4y-2x=3\Rightarrow y=\frac{1}{2}x+\frac{3}{4}$

so the slope is $\frac{1}{2}$ .

Since the slope of the second equation is not equal to 3, then the lines are not perpendicular.

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Question

Figure NOT drawn to scale.

Refer to the above figure.

True or false: $m \perp t$

Statement 1: $\angle 4$ is a right angle.

Statement 2: $\angle 3$ and $\angle 5$ are supplementary.

Answer

Statement 1 alone establishes by definition that $l\perp t$ , but does not establish any relationship between and .

By Statement 2 alone, since same-side interior angles are supplementary, , 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. If two lines are parallel, then any line in their plane perpendicular to one must be perpendicular to the other. and $l\perp t$ , so it can be established that $m \perp t$ .

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Question

Refer to the above figure. . True or false: $m \perp t$

Statement 1: $\angle 3 \cong \angle 5$

Statement 2: $\angle 2$ and $\angle 6$ are supplementary.

Answer

If transversal crosses two parallel lines and , then same-side interior angles are supplementary, so $\angle 3$ and $\angle 5$ are supplementary angles. Also, corresponding angles are congruent, so $\angle 2 \cong \angle 6$ .

By Statement 1 alone, angles $\angle 3$ and $\angle 5$ are congruent as well as supplementary; by Statement 2 alone, $\angle 2$ and $\angle 6$ are also supplementary as well as congruent. Two angles that are both supplementary and congruent are both right angles, so from either statement alone, and intersect at right angles, so, consequently, $m \perp t$ .

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Question

Find the equation of the line that is perpendicular to the following equation and passes through the point .

$y=-\frac{x}{2}+14$

Answer

To solve this equation, we want to begin by recalling how to find the slope of a perpendicular line. In this case, our original line is modeled by the following:

$y=-\frac{x}{2}+14$

To find the slope of any line perpendicular to the above equation, we simply need to take the reciprocal of the first slope, and then change its sign. Our original slope is $-\frac{1}{2}$ , so

$m=-\frac{1}{2}$

becomes

.

If we flip $\frac{1}{2}$ , we get , and the opposite sign of a negative is a positive; hence, our slope is positive .

So, we know our perpendicular line should look something like this:

However, we need to find out what (our -intercept) is in order to complete our equation. To do so, we need to plug in the ordered pair we received in the question, , and solve for :

So, by putting everything together, we get our final equation:

This equation satisfies the conditions of being perpendicular to our initial equation and passing through .

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Question

Which of the following lines is perpendicular to $y=-\frac{3}{4}x+7$ ?

Answer

In order for a line to be perpendicular to another line defined by the equation , the slope of line must be a negative reciprocal of the slope of line . Since line 's slope is in the slope-intercept equation above, line 's slope would therefore be $-\frac{1}{m}$ .

In this instance, $m=-\frac{3}{4}$ , so $-\frac{1}{m}=-\frac{1}{-\frac{3}{4}}=\frac{4}{3}$ . Therefore, the correct solution is $y=\frac{4}{3}x+7$ .

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Question

A given line has a slope of . What is the slope of any line perpendicular to ?

Answer

In order for a line to be perpendicular to another line defined by the equation , the slope of line must be a negative reciprocal of the slope of line . Since line 's slope is in the slope-intercept equation above, line 's slope would therefore be $-\frac{1}{m}$ .

Given that we have a line with a slope , we can therefore conclude that any perpendicular line would have a slope $-\frac{1}{m}=-\frac{1}{6}$ .

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