Question

the diagram shows a regular polygon.
what is the value of x?
write your answer as an integer or as a decimal rounded to the nearest tenth.
x = °

Explanation:

Step1: Identify the polygon type

The diagram shows a regular quadrilateral (square or rhombus, but regular, so square). A regular quadrilateral has 4 sides.

Step2: Find the measure of an interior angle

For a regular polygon, the formula for the measure of an interior angle is $\frac{(n - 2)\times180^{\circ}}{n}$, where $n$ is the number of sides. For $n = 4$, we have $\frac{(4 - 2)\times180^{\circ}}{4}=\frac{2\times180^{\circ}}{4}=90^{\circ}$.

Step3: Find the exterior angle

An interior angle and its corresponding exterior angle are supplementary (they add up to $180^{\circ}$). So, $x = 180^{\circ}- 90^{\circ}=90^{\circ}$? Wait, no, wait. Wait, a regular quadrilateral (square) has exterior angles. Wait, no, the formula for exterior angle of a regular polygon is $\frac{360^{\circ}}{n}$. For $n = 4$, $\frac{360^{\circ}}{4}=90^{\circ}$? Wait, no, that can't be. Wait, no, the diagram: the angle $x$ is an exterior angle? Wait, the regular polygon here is a quadrilateral (4 sides). The sum of exterior angles of any polygon is $360^{\circ}$. For a regular polygon, each exterior angle is $\frac{360^{\circ}}{n}$. So for $n = 4$, each exterior angle is $\frac{360^{\circ}}{4}=90^{\circ}$? Wait, but that would mean the interior angle is $90^{\circ}$, which is correct for a square. Wait, but maybe I misread the diagram. Wait, the diagram shows a regular polygon, which looks like a square (4 sides). So the exterior angle $x$: since in a regular polygon, exterior angle = $\frac{360}{n}$. So $n = 4$, so $x=\frac{360}{4}=90$? Wait, no, that seems off. Wait, no, maybe the polygon is a different one? Wait, no, the diagram is a regular quadrilateral. Wait, maybe I made a mistake. Wait, no, the sum of exterior angles of any convex polygon is $360^{\circ}$. So for a regular polygon with $n$ sides, each exterior angle is $\frac{360}{n}$. So if $n = 4$, then each exterior angle is $90^{\circ}$. So $x = 90$.

Wait, but let's check again. The interior angle of a regular quadrilateral is $90^{\circ}$, so the exterior angle (supplementary to interior angle) is $180 - 90 = 90^{\circ}$. So that's correct.

Answer:

$90$