Question

determine if the statement is always, sometimes or never true.
an exterior angle of a triangle is 180 degrees.
never
always
sometimes

Explanation:

Step1: Recall exterior angle definition

An exterior angle of a triangle is formed by one side of the triangle and the extension of an adjacent side. By the linear pair postulate, an exterior angle and its adjacent interior angle are supplementary (sum to \(180^\circ\)), but the exterior angle itself is equal to the sum of the two non - adjacent interior angles (exterior angle theorem). The measure of an interior angle of a triangle is between \(0^\circ\) and \(180^\circ\) (excluding \(0^\circ\) and \(180^\circ\) for a non - degenerate triangle). So the exterior angle, being equal to the sum of two positive interior angles (each less than \(180^\circ\)) and also supplementary to an interior angle, must be less than \(180^\circ\) (since the interior angle is greater than \(0^\circ\)). For example, in a right - angled triangle with interior angles \(90^\circ\), \(30^\circ\), and \(60^\circ\), the exterior angle adjacent to the \(90^\circ\) angle is \(90^\circ\), adjacent to \(30^\circ\) is \(150^\circ\), and adjacent to \(60^\circ\) is \(120^\circ\), none of which are \(180^\circ\).

Step2: Conclusion about the statement

Since an exterior angle of a triangle is always supplementary to an interior angle (so it and the interior angle add to \(180^\circ\)), but the exterior angle itself cannot be \(180^\circ\) (because the interior angle would then be \(0^\circ\), which is not possible for a triangle). So the statement "An exterior angle of a triangle is \(180\) degrees" is never true.

Answer:

never