Question

a side of the triangle below has been extended to form an exterior angle of 72°. find the value of x.
triangle image with one angle 55°, another angle x°, and an exterior angle 72°

Explanation:

Step1: Recall the exterior angle theorem

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. But also, a straight angle is \(180^{\circ}\), so the interior angle adjacent to the exterior angle of \(72^{\circ}\) and \(x^{\circ}\) satisfies \(x + 72=180\)? No, wait, the exterior angle theorem: the exterior angle is equal to the sum of the two remote interior angles. Wait, in this case, the exterior angle is \(72^{\circ}\)? Wait, no, looking at the diagram, the angle of \(72^{\circ}\) and \(x^{\circ}\) are adjacent and form a linear pair? Wait, no, the triangle has an angle of \(55^{\circ}\), and we need to find \(x\) such that the exterior angle (let's correct: the angle formed by extending a side is an exterior angle, and the two angles \(x\) and \(55^{\circ}\) should be related to the exterior angle. Wait, the correct approach: the exterior angle is equal to the sum of the two non - adjacent interior angles. Wait, maybe I misread. Let's see: the angle of \(72^{\circ}\) is adjacent to \(x\), so \(x+72 = 180\)? No, that would be a linear pair, but then the other angle is \(55^{\circ}\). Wait, no, the exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two remote (non - adjacent) interior angles. Wait, maybe the exterior angle here is equal to \(x + 55\)? Wait, no, the diagram: the triangle has angles \(x\), \(55^{\circ}\), and the angle adjacent to \(72^{\circ}\). Wait, the angle adjacent to \(72^{\circ}\) and \(x\) is a straight angle, so \(x + 72=\) the angle adjacent? No, let's start over.

The sum of the interior angles of a triangle is \(180^{\circ}\). Also, the angle adjacent to \(72^{\circ}\) (let's call it \(y\)) and \(72^{\circ}\) form a linear pair, so \(y+72 = 180\), so \(y = 180 - 72=108\)? No, that can't be, because the triangle has a \(55^{\circ}\) angle. Wait, no, I think I made a mistake. The correct exterior angle theorem: the exterior angle is equal to the sum of the two non - adjacent interior angles. So if the exterior angle is \(72^{\circ}\), then \(72=x + 55\)? Wait, that would make \(x=72 - 55 = 17\)? No, that doesn't seem right. Wait, no, the angle of \(72^{\circ}\) is adjacent to \(x\), so \(x+72\) is not the case. Wait, let's look at the diagram again. The triangle has angles: \(x\), \(55^{\circ}\), and the angle that is supplementary to \(72^{\circ}\). Wait, the angle supplementary to \(72^{\circ}\) is \(180 - 72=108^{\circ}\). Then, since the sum of angles in a triangle is \(180^{\circ}\), we have \(x + 55+108 = 180\)? No, that would be \(x+163 = 180\), \(x = 17\). Wait, that can't be. Wait, no, the exterior angle is equal to the sum of the two non - adjacent interior angles. So the exterior angle (the \(72^{\circ}\) angle) is equal to the sum of \(x\) and \(55^{\circ}\)? Wait, no, that would mean \(x+55 = 72\), so \(x=72 - 55 = 17\). Wait, that seems correct. Let's verify:

If \(x = 17^{\circ}\), and the other angle (adjacent to \(72^{\circ}\)) is \(180 - 72=108^{\circ}\), then the sum of angles in the triangle is \(17+55 + 108=180\) (since \(17+55 = 72\), \(72 + 108=180\)). Yes, that works. So the correct approach is:

The exterior angle ( \(72^{\circ}\)) is equal to the sum of the two non - adjacent interior angles (\(x\) and \(55^{\circ}\)). So we have the equation \(x + 55=72\).

Step2: Solve for \(x\)

Subtract \(55\) from both sides of the equation \(x + 55=72\).
\(x=72 - 55\)
\(x = 17\)

Answer:

\(x = 17\)