Question

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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 = \boxed{}^\circ

Explanation:

Step1: Identify the polygon type

The diagram shows a regular quadrilateral (a square or regular rhombus, but since it's regular, all interior angles are equal). For a regular polygon with \( n \) sides, the measure of each interior angle \( I \) is given by \( I=\frac{(n - 2)\times180^{\circ}}{n} \). For a quadrilateral, \( n = 4 \).

Step2: Calculate the interior angle

Substitute \( n = 4 \) into the formula: \( I=\frac{(4 - 2)\times180^{\circ}}{4}=\frac{2\times180^{\circ}}{4}=90^{\circ} \).

Step3: Find the exterior angle \( x \)

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, the diagram: wait, maybe I misidentified. Wait, the regular polygon here—wait, the diagram is a regular quadrilateral? Wait, no, wait, the regular polygon in the diagram: wait, the figure is a regular quadrilateral? Wait, no, maybe it's a regular polygon with 4 sides? Wait, no, wait, the exterior angle of a regular polygon is \( \frac{360^{\circ}}{n} \). For \( n = 4 \), exterior angle is \( \frac{360^{\circ}}{4}=90^{\circ} \)? Wait, no, that's not right. Wait, no, the formula for exterior angle of a regular polygon is \( \frac{360^{\circ}}{n} \). So for a square (regular quadrilateral, \( n = 4 \)), each exterior angle is \( \frac{360}{4}=90^{\circ} \)? Wait, no, that can't be. Wait, no, interior angle of square is \( 90^{\circ} \), so exterior angle is \( 180 - 90 = 90^{\circ} \). Wait, but that's correct? Wait, no, wait, for a regular polygon, the sum of exterior angles is always \( 360^{\circ} \), so each exterior angle is \( \frac{360}{n} \). So for \( n = 4 \), \( \frac{360}{4}=90^{\circ} \). So \( x = 90^{\circ} \)? Wait, but the diagram: the angle \( x \) is an exterior angle. So yes, for a regular quadrilateral (square), each exterior angle is \( 90^{\circ} \). Wait, but that seems odd, but mathematically, it's correct. Wait, no, wait, no—wait, no, the interior angle of a square is \( 90^{\circ} \), so exterior angle is \( 180 - 90 = 90^{\circ} \), and also, since sum of exterior angles is \( 360^{\circ} \), \( 4 \times 90 = 360 \), which checks out. So \( x = 90 \).

Answer:

\( 90 \)