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?

At an intersection, vertical angles are opposite and equal, while a linear pair consists of neighboring angles that add to $180^\circ$ . Use the given obtuse-to-acute ratio to write both measures in terms of the acute angle, then use a Desmos regression to find them. Check every type of two-angle sum. The constant in an angle expression isn't a separate angle.

Hints

Hint 1
The ratio compares the obtuse angle to the acute angle. If the acute measure is $9x-60$ , multiplying that entire expression by $\frac{5}{4}$ gives the obtuse measure.
Hint 2
An acute angle and an obtuse angle beside it form a linear pair: together they make a straight angle of $180^\circ$ . What equation combines that total with the ratio?
Hint 3
Vertical angles sit opposite each other and have equal measures. Once you know the acute and obtuse measures, check all three pair types: two acute, one of each, and two obtuse.

Step-by-step

Find both angles, then list the sums
Step 1
Express the obtuse angle
The given acute angle measures $9x-60$ degrees. The ratio is obtuse to acute, $\frac{5}{4}$ , so an obtuse angle measures $\frac{5}{4}(9x-60)$ degrees. The $60$ is part of the acute angle's expression, not another angle.
Step 2
Use the straight angle
An acute and an obtuse angle beside each other form a linear pair: they fill a straight line, so their measures add to $180^\circ$ . That gives the equation:
$\left(9x-60\right)+\frac{5}{4}\left(9x-60\right)=180.$
Step 3
Solve for the input in Desmos
Type the equation with $x$ changed to $x_1$ and $=$ changed to $\sim$ . Desmos uses $\sim$ to solve for $x_1$ and shows $x_1\approx15.556$ under PARAMETERS. That is an input, not an angle measure.
Step 4
Read the acute measure
Type $9x_1-60$ . Desmos uses the value it found for $x_1$ and prints $80$ . So the given acute angle measures $80^\circ$ ; its opposite vertical angle also measures $80^\circ$ .
Step 5
Read the obtuse measure
Type $\frac{5}{4}(9x_1-60)$ . Desmos prints $100$ , so each obtuse angle measures $100^\circ$ . This fits the straight line: $80^\circ+100^\circ=180^\circ$ .
Step 6
Check every possible pair
The only pair types are two acute angles, one of each, or two obtuse angles. Add their measures:
$\begin{aligned}80+80&=160\\[1.4em]80+100&=180\\[1.4em]100+100&=200.\end{aligned}$
So $140^\circ$ cannot be the sum of any two of the four angles. Choice C.

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SAT Geometry and TrigonometryBeginnerCoreUse line and angle relationships

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

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