Question 92·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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A regular polygon has an interior angle measure of $(10x+20)^\circ$ . Label three consecutive vertices $A$ , $B$ , and $C$ . A diagonal is drawn from $A$ to $C$ (so the diagonal skips exactly one vertex, $B$ ). The angle between diagonal $AC$ and side $AB$ is $(2x-18)^\circ$ . Which choice is the number of sides of the polygon?

A diagonal that skips one vertex of a regular polygon forms an isosceles triangle: the two polygon sides in that triangle are equal. Match the equal angles, then use the triangle’s $180^\circ$ total to find the polygon’s interior angle. Its exterior angle completes $180^\circ$ , and the exterior angles total $360^\circ$ . Don’t mistake the solved variable for the number of sides.

Hints

Hint 1
A regular polygon has equal sides, so $AB=BC$ . In triangle $ABC$ , which two angles lie across from those equal sides? They must have equal measures.
Hint 2
The polygon’s full interior angle is at $B$ , between sides $BA$ and $BC$ . Triangle $ABC$ contains that angle and two equal angles beside the diagonal. What do the three angles add up to?
Hint 3
An exterior angle is the turn outside the polygon at a vertex. It adds to the interior angle to make $180^\circ$ . All the exterior turns add to $360^\circ$ , so which angle should divide $360^\circ$ ?

Step-by-step

Use the triangle, then the exterior angle
Step 1
Match the triangle’s equal angles
Because the polygon is regular, its sides $AB$ and $BC$ are equal. Equal sides face equal angles, so triangle $ABC$ has equal angles at $A$ and $C$ : $m\angle BAC=m\angle ACB=(2x-18)^\circ$ . The diagonal-to-side angle appears twice in this triangle.
Step 2
Write the triangle-angle equation
At $B$ , $BA$ and $BC$ are polygon sides, so $\angle ABC$ is the full interior angle, $(10x+20)^\circ$ . A triangle’s angles add to $180^\circ$ , so: $2(2x-18)+(10x+20)=180$ . Don’t put the interior angle at $A$ : one side of that angle is the diagonal.
Step 3
Solve for the variable
Type the triangle equation in Desmos, changing $x$ to $x_1$ and $=$ to $\sim$ so Desmos solves for the unknown value. Under PARAMETERS, it shows $x_1=14$ . That’s $x$ , not the side count.
Step 4
Find the polygon’s interior angle
The question gives the interior angle as $(10x+20)^\circ$ . Type $10x_1+20$ below the regression; Desmos shows $160$ . So each interior angle measures $160^\circ$ .
Step 5
Find one exterior angle
An exterior angle and its neighboring interior angle make a straight $180^\circ$ angle. Subtract the interior angle: $180-160=20$ . Each outside turn is $20^\circ$ .
Step 6
Find the number of sides
The exterior turns of a polygon add to $360^\circ$ . Divide by the turn at each vertex: $n=\frac{360}{20}=18$ . The polygon has $18$ sides. Choice D.

Lessons that teach this

SAT Geometry and TrigonometryIntermediateCoreUse triangle angle theorems

Question 92·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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