Super-Hard SAT Math Question 69 | Free SAT Prep | Aniko

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Question 69·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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Two lines intersect at exactly one point, forming two acute angles and two obtuse angles. One of the acute angles measures $(9x - 60)^\circ$ , where $x$ is a real number. The ratio of the measure of an obtuse angle to the measure of an acute angle is $\frac{5}{4}$ . Which choice could NOT be the sum of the measures of any two of the four angles formed by the intersecting lines?

For intersecting lines, define the acute angle as $a$ and the obtuse angle as its supplement $b=180-a$ . Use any extra relationship (like a ratio) to solve for $a$ (and thus $b$ ). Then remember there are only three distinct sums you can form from two of the four angles: $2a$ , $2b$ , and $180^\circ$ . Compare those to the answer choices.

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Use supplements

If the acute angle is $a$ , the obtuse angle must be $180-a$ .

Turn the ratio into an equation

Use $\dfrac{180-a}{a}=\dfrac{5}{4}$ to solve for $a$ .

There are only three different sums

From $a,a,b,b$ , the only distinct sums of two angles are $2a$ , $2b$ , and $a+b$ .

Desmos Guide

Solve for the acute angle using the ratio

In Desmos, let a be the acute angle measure. Enter the equation 4(180-a)=5a and solve for a (Desmos will show a=80).

Compute the obtuse angle

Enter b=180-a to get b=100.

Generate the only possible sums

Enter 2a, 2b, and a+b to see the three possible sums: 160, 200, and 180.

Compare to the choices

The correct choice is the option that is not one of 160, 180, or 200.

Step-by-step Explanation

Represent the two distinct angle measures

Let the acute angle be

a = 9x - 60.

Its adjacent (obtuse) angle is supplementary, so

b = 180 - a.

Use the given ratio to find $a$

The problem states

\frac{b}{a} = \frac{5}{4}.

Substitute $b=180-a$ :

\begin{aligned} \frac{180-a}{a} &= \frac{5}{4} \\ 4(180-a) &= 5a \\ 720 - 4a &= 5a \\ 720 &= 9a \\ a &= 80. \end{aligned}

Then $b=180-a=100$ . (Finding $x$ is not necessary to answer the question.)

List all possible sums of two of the four angles

The four angles are $a, a, b, b$ .

So any sum of two angles must be one of:

$a+a = 2a = 160^\circ$
$b+b = 2b = 200^\circ$
$a+b = 180^\circ$

Compare with the choices

The possible sums are $160^\circ$ , $180^\circ$ , and $200^\circ$ .

The only listed value that is not possible is $140^\circ$ .

Strategy

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation