Super-Hard SAT Math Question 19 | Free SAT Prep | Aniko

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Question 19·200 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. The measure of one of the acute angles is $\left(\frac{5}{3}x - 40\right)^\circ$ .

A third line passes through the intersection point and bisects each of the obtuse angles formed by the two original lines. A fourth line passes through the intersection point and bisects each of the acute angles formed by the two original lines.

Which choice is NOT equivalent (as an expression in $x$ ) to the sum of the measures of any three of the eight angles formed by the four lines?

For intersecting lines, reduce everything to one acute angle $E$ and its supplement $S=180-E$ . When bisectors are added, immediately rewrite the diagram in terms of the resulting small angles (here, only $E/2$ and $S/2$ ). For “sum of $k$ angles” questions, count how many of each type can be chosen (combinations of $E/2$ and $S/2$ ), generate the limited list of possible sum-forms, then convert each to an expression in $x$ and compare by equivalence.

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Name the acute angle

Let the given acute angle be $E$ . What is the measure of the obtuse angle $S$ in terms of $E$ ?

Use both bisectors

After bisecting, what are the measures of the small angles created from $E$ and from $S$ ?

Count how many of each type you can pick

If each small angle is either $E/2$ or $S/2$ , then the sum of any three angles is $a\cdot(E/2)+b\cdot(S/2)$ with $a+b=3$ . List the possibilities for $a$ .

Convert each possible sum into $x$

Substitute $S=180-E$ and $E=\left(\frac{5}{3}x-40\right)$ to turn each possible sum into an expression in $x$ .

Desmos Guide

Define $E$ and $S$

In Desmos, enter

$E=(5/3)x-40$
$S=180-E$

Define the four possible three-angle sums

Enter

$A=3E/2$
$B=E+S/2$
$C=E/2+S$
$D=3S/2$

Enter the four answer choices

Enter each answer choice (without the degree symbol), for example:

$f_1=(5/2)x-60$
$f_2=70+(5/6)x$
$f_3=200-(5/6)x$
$f_4=330+(5/2)x$

Identify which choice never matches

A correct (allowable) expression will have a graph that coincides with one of $A$ , $B$ , $C$ , or $D$ . The option whose graph does not coincide with any of these is the correct answer.

Step-by-step Explanation

Define the acute and obtuse angles

Let

E=\left(\frac{5}{3}x-40\right)^\circ.

The obtuse angle formed by the original two lines is supplementary to $E$ , so

S=(180-E)^\circ.

Use the fact that both angle types are bisected

The obtuse-angle bisector splits each obtuse angle $S$ into two angles of measure $S/2$ .

The acute-angle bisector splits each acute angle $E$ into two angles of measure $E/2$ .

So among the eight angles formed, there are:

four angles of measure $E/2$
four angles of measure $S/2$

Classify all possible sums of three angles

Any sum of three of the eight angles has the form

a\cdot\frac{E}{2}+b\cdot\frac{S}{2}

where $a$ is how many $E/2$ angles are chosen and $b$ is how many $S/2$ angles are chosen, with $a+b=3$ .

Thus the only possibilities are:

$3\cdot\frac{E}{2}=\frac{3E}{2}$
$2\cdot\frac{E}{2}+1\cdot\frac{S}{2}=E+\frac{S}{2}$
$1\cdot\frac{E}{2}+2\cdot\frac{S}{2}=\frac{E}{2}+S$
$3\cdot\frac{S}{2}=\frac{3S}{2}$

Write each possible sum as an expression in $x$

Substitute $S=180-E$ and $E=\left(\frac{5}{3}x-40\right)$ .

\frac{3E}{2}=\frac{3}{2}\left(\frac{5}{3}x-40\right)=\frac{5}{2}x-60

E+\frac{S}{2}=E+\frac{180-E}{2}=90+\frac{E}{2}=90+\frac{1}{2}\left(\frac{5}{3}x-40\right)=70+\frac{5}{6}x

\frac{E}{2}+S=\frac{E}{2}+180-E=180-\frac{E}{2}=180-\frac{1}{2}\left(\frac{5}{3}x-40\right)=200-\frac{5}{6}x

\frac{3S}{2}=\frac{3}{2}(180-E)=270-\frac{3E}{2}=270-\left(\frac{5}{2}x-60\right)=330-\frac{5}{2}x

Match to the choices

Three of the choices match allowable sums: $\left(\frac{5}{2}x-60\right)^\circ$ , $\left(70+\frac{5}{6}x\right)^\circ$ , and $\left(200-\frac{5}{6}x\right)^\circ$ .

The only remaining choice, $\left(330+\frac{5}{2}x\right)^\circ$ , does not match any allowable three-angle sum, so it is the answer.

Strategy

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation