Question

question a side of the triangle below has been extended to form an exterior angle of 128°. find the value of x.

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. Let's denote the interior angle adjacent to the \(128^{\circ}\) exterior angle as \(x\) and the other non - adjacent interior angle as \(106^{\circ}\). We know that the exterior angle (\(128^{\circ}\)) is equal to the sum of the two non - adjacent interior angles (\(x\) and \(106^{\circ}\))? Wait, no, actually, the exterior angle and its adjacent interior angle are supplementary. Wait, let's correct that. The angle adjacent to the \(128^{\circ}\) exterior angle is \(x\), so \(x + 128^{\circ}=180^{\circ}\)? No, that's not right. Wait, the triangle has an angle of \(106^{\circ}\), and the exterior angle is \(128^{\circ}\). By the exterior angle theorem, the exterior angle is equal to the sum of the two remote interior angles. Wait, let's look at the triangle. The exterior angle is \(128^{\circ}\), one of the interior angles is \(106^{\circ}\), and the other interior angle is \(x\). So according to the exterior angle theorem, \(128^{\circ}=106^{\circ}+x\)? Wait, no, that would give \(x = 128 - 106=22\)? Wait, no, maybe I mixed up. Wait, the exterior angle and the adjacent interior angle are supplementary, so the adjacent interior angle is \(180 - 128 = 52^{\circ}\)? No, that can't be. Wait, let's start over.

In a triangle, the sum of the interior angles is \(180^{\circ}\). Let's denote the three interior angles as \(A = 106^{\circ}\), \(B=x\), and \(C\) (the angle adjacent to the exterior angle of \(128^{\circ}\)). We know that \(C + 128^{\circ}=180^{\circ}\) (since they are supplementary, forming a linear pair), so \(C=180 - 128 = 52^{\circ}\). Then, since the sum of the interior angles of a triangle is \(180^{\circ}\), we have \(A + B+C=180^{\circ}\). Substituting \(A = 106^{\circ}\) and \(C = 52^{\circ}\) into the equation: \(106+x + 52=180\). Then \(x+158 = 180\), so \(x=180 - 158=22\)? Wait, no, that's not correct. Wait, maybe the exterior angle is equal to the sum of the two non - adjacent interior angles. So the exterior angle (\(128^{\circ}\)) is equal to the sum of the two non - adjacent interior angles, which are \(106^{\circ}\) and \(x\). So \(128=106 + x\), then \(x=128 - 106 = 22\)? Wait, but let's check with the supplementary angle. If the exterior angle is \(128^{\circ}\), the adjacent interior angle is \(180 - 128 = 52^{\circ}\). Then the sum of the interior angles of the triangle is \(106 + x+52=180\), so \(x=180-(106 + 52)=180 - 158 = 22\). Yes, that's correct.

Step2: Solve for \(x\)

Using the exterior angle theorem: The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So we have the equation \(128^{\circ}=106^{\circ}+x\).

To solve for \(x\), we subtract \(106^{\circ}\) from both sides of the equation:

\(x=128^{\circ}- 106^{\circ}\)

\(x = 22^{\circ}\)

Answer:

\(x = 22\)