A regular polygon has an interior angle measure of . Label three consecutive vertices , , and . A diagonal is drawn from to (so the diagonal skips exactly one vertex, ). The angle between diagonal and side is . Which choice is the number of sides of the polygon?

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Super-Hard SAT Math Question 92 | Free SAT Prep | Aniko

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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?

When a diagonal in a regular polygon connects a vertex to the vertex two away, look at the triangle made by the two consecutive sides and that diagonal. Because consecutive sides are equal, the triangle is isosceles: the diagonal-side angle is a base angle, so it equals $\frac{180-\text{(interior angle)}}{2}$ . After you solve for the interior angle, find the exterior angle by subtracting from $180^\circ$ , then use $\frac{360^\circ}{n}$ to get $n$ .

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Use an isosceles triangle

If $A$ , $B$ , and $C$ are consecutive vertices and you draw diagonal $AC$ , what can you say about triangle $ABC$ in a regular polygon?

Connect base angles to the interior angle

In $\triangle ABC$ , the angle at $B$ is the polygon’s interior angle. How do you express the base angle at $A$ in terms of that interior angle?

Finish with exterior angles

After you find the interior angle measure, subtract from $180^\circ$ to get the exterior angle, then use $\frac{360^\circ}{n}$ to get $n$ .

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Graph the two expressions for the same angle

Enter

$y=(180-(10x+20))\div 2$
$y=2x-18$

The intersection’s $x$ -value is the solution for $x$ .

Compute the interior and exterior angles

Using that $x$ -value, compute the interior angle $10x+20$ . Then compute the exterior angle $180-(10x+20)$ .

Compute the number of sides

Enter $n=360\div(180-(10x+20))$ using the same $x$ -value. The resulting $n$ matches the correct answer choice.

Step-by-step Explanation

Relate the diagonal angle to the interior angle

Let $A$ , $B$ , and $C$ be three consecutive vertices, and draw diagonal $AC$ . Then $AB=BC$ (regular polygon), so $\triangle ABC$ is isosceles.

The angle at $B$ is the polygon’s interior angle:

\angle ABC = (10x+20)^\circ.

Therefore the base angles at $A$ and $C$ are equal and sum to $180^\circ-(10x+20)^\circ$ , so each base angle is

\frac{180-(10x+20)}{2}.

Set up and solve the equation for $x$

The angle between diagonal $AC$ and side $AB$ is the base angle at $A$ , which is given as $(2x-18)^\circ$ . Set them equal:

\frac{180-(10x+20)}{2} = 2x-18

Solve:

\begin{aligned} \frac{160-10x}{2} &= 2x-18 \\ 80-5x &= 2x-18 \\ 98 &= 7x \\ x &= 14 \end{aligned}

Find the exterior angle

Compute the interior angle:

10x+20 = 10(14)+20 = 160^\circ.

For a polygon, an interior angle and its (non-reflex) exterior angle are supplementary, so the exterior angle is

180-160 = 20^\circ.

Use the exterior-angle rule to find the number of sides

In a regular $n$ -gon, each exterior angle equals $\frac{360^\circ}{n}$ .

\frac{360}{n} = 20

which gives $n=18$ .

Therefore, the polygon has 18 sides.

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