Question

a side of the triangle below has been extended to form an exterior angle of 70°. 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. But also, we know that a straight angle is \(180^{\circ}\), and we can use the fact that the sum of angles in a triangle is \(180^{\circ}\) or the linear pair property. First, let's consider the linear pair: the angle adjacent to the \(70^{\circ}\) exterior angle and \(x\) forms a linear pair? Wait, no. Wait, the exterior angle is \(70^{\circ}\), and we have a triangle with one angle \(50^{\circ}\), another angle \(x\), and the third angle is supplementary to \(70^{\circ}\) (since they form a linear pair). Wait, the angle adjacent to the \(70^{\circ}\) exterior angle is \(180 - 70=110^{\circ}\)? No, that's not right. Wait, no. Wait, the exterior angle theorem: the exterior angle is equal to the sum of the two remote interior angles. Wait, the exterior angle here is \(70^{\circ}\)? Wait, no, maybe I misread. Wait, the diagram: a side is extended to form an exterior angle of \(70^{\circ}\). So the interior angle adjacent to the exterior angle is \(180 - 70 = 110^{\circ}\)? No, that can't be, because the triangle has an angle of \(50^{\circ}\) and \(x\), and the sum of angles in a triangle is \(180^{\circ}\). Wait, no, let's correct. The exterior angle is equal to the sum of the two non - adjacent interior angles. Wait, maybe the exterior angle is \(70^{\circ}\), and the two non - adjacent interior angles are \(50^{\circ}\) and \(x\)? Wait, no, that would mean \(70=50 + x\), so \(x = 20\)? Wait, no, that doesn't seem right. Wait, no, maybe the exterior angle is formed by extending a side, so the interior angle at that vertex is \(180 - 70=110^{\circ}\), and then the sum of angles in a triangle is \(180^{\circ}\), so \(50 + x+110 = 180\)? No, that would give \(x = 20\), but that contradicts. Wait, no, I think I made a mistake. Wait, let's start over.

The sum of the interior angles of a triangle is \(180^{\circ}\). Also, a linear pair of angles (adjacent angles on a straight line) sum to \(180^{\circ}\). Let's denote the angle adjacent to the \(70^{\circ}\) exterior angle as \(y\). Then \(y + 70=180\), so \(y = 110^{\circ}\). Now, in the triangle, we have angles \(50^{\circ}\), \(x\), and \(y\)? No, that can't be, because \(50 + x+110=160 + x\), which would be more than \(180\) if \(x>0\). So that's wrong. Wait, maybe the exterior angle is \(70^{\circ}\), and the two non - adjacent interior angles are \(x\) and \(50^{\circ}\). So by exterior angle theorem, \(70=x + 50\). Then \(x=70 - 50=20\)? Wait, that seems too simple. Wait, let's check the diagram again. The triangle has an angle of \(50^{\circ}\), angle \(x\), and the exterior angle is \(70^{\circ}\) which is equal to the sum of the two non - adjacent interior angles. So if the exterior angle is \(70^{\circ}\), and one non - adjacent interior angle is \(50^{\circ}\), then the other non - adjacent interior angle \(x\) satisfies \(x + 50=70\). So \(x=70 - 50 = 20\). Wait, but let's confirm with the sum of angles in a triangle. The third angle (adjacent to the exterior angle) is \(180 - 70 = 110^{\circ}\)? No, that can't be, because \(50+20 + 110=180\), yes! Wait, \(50 + x+(180 - 70)=180\). So \(50+x + 110=180\), so \(x=180-(50 + 110)=20\). Yes, that works.

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. Let the exterior angle be \(70^{\circ}\), one non - adjacent interior angle be \(50^{\circ}\), and the other non - adjacent interior angle be \(x\). Then:
\(70=50 + x\)
Subtract \(50\) from both sides:
\(x=70 - 50\)
\(x = 20\)

Answer:

\(20\)