Question

determine if the statement is always, sometimes or never true. the measure of an exterior angle at the vertex of a polygon equals the measure of the adjacent interior angle. never sometimes always

Explanation:

Step1: Recall the relationship between exterior and adjacent interior angles

An exterior angle and its adjacent interior angle at a vertex of a polygon are supplementary, meaning their sum is \(180^\circ\) (by the linear pair postulate, since they form a straight line). So, if we let the measure of the interior angle be \(I\) and the exterior angle be \(E\), we have \(I + E=180^\circ\).

Step2: Analyze when \(I = E\)

If \(I = E\), then substituting into the equation \(I + E = 180^\circ\), we get \(I+I=180^\circ\), which simplifies to \(2I = 180^\circ\), so \(I = 90^\circ\) and \(E = 90^\circ\). This occurs in specific cases, like in a square (a regular quadrilateral) where each interior angle is \(90^\circ\) and each exterior angle is also \(90^\circ\) (since the sum of exterior angles of any polygon is \(360^\circ\), and for a square, \(360^\circ\div4 = 90^\circ\)). However, for other polygons, like a regular triangle (equilateral triangle), the interior angle is \(60^\circ\) and the exterior angle is \(120^\circ\) (since \(180 - 60=120\)), so they are not equal. So the statement is true in some cases (when the interior angle is \(90^\circ\)) and false in others.

Answer:

sometimes