Question

a side of the triangle below has been extended to form an exterior angle of 71°. find the value of x.
answer attempt 2 out of 2
x =

Explanation:

Step1: Recall the exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. Also, we know that a straight angle is \(180^{\circ}\), but here we can use the fact that in a triangle, the sum of an interior angle and its adjacent exterior angle is equal to the sum of the other two interior angles. Wait, more simply, we know that the exterior angle is equal to the sum of the two remote interior angles. Wait, no, let's look at the angles. We have an exterior angle of \(71^{\circ}\), one interior angle of \(49^{\circ}\), and we need to find \(x\). Wait, actually, the exterior angle and the angle \(x\) and \(49^{\circ}\): Wait, no, the exterior angle is formed by extending a side, so the angle adjacent to the exterior angle (which is \(x\)) and the exterior angle \(71^{\circ}\) form a linear pair? No, wait, no. Wait, the exterior angle theorem: the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. So here, the exterior angle is \(71^{\circ}\), one non - adjacent interior angle is \(49^{\circ}\), and the other non - adjacent interior angle is \(x\)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, actually, the angle \(x\) and the exterior angle \(71^{\circ}\) and the angle \(49^{\circ}\): Wait, let's think again. The sum of the interior angles of a triangle is \(180^{\circ}\), and the exterior angle is equal to the sum of the two remote interior angles. Wait, no, the correct approach is: the exterior angle is equal to the sum of the two non - adjacent interior angles. So if the exterior angle is \(71^{\circ}\), and one non - adjacent interior angle is \(49^{\circ}\), and the other non - adjacent interior angle is \(x\), then \(71^{\circ}=49^{\circ}+x\)? Wait, no, that would mean \(x = 71 - 49=22\)? Wait, no, that's not right. Wait, no, maybe the exterior angle and \(x\) are related as follows: the exterior angle is equal to the sum of the two interior angles that are not adjacent to it. Wait, let's look at the diagram. The triangle has an angle of \(49^{\circ}\), an angle \(x\), and the exterior angle of \(71^{\circ}\) is adjacent to the angle that is supplementary to \(x\)? No, wait, the exterior angle and \(x\) and \(49^{\circ}\): Wait, the sum of the interior angles of a triangle is \(180^{\circ}\), and the exterior angle is equal to the sum of the two opposite interior angles. So if the exterior angle is \(71^{\circ}\), and one opposite interior angle is \(49^{\circ}\), then the other opposite interior angle \(x\) can be found by \(x=71 - 49\)? Wait, no, that would be if the exterior angle is equal to the sum of the two non - adjacent angles. Wait, let's take an example. Suppose we have a triangle, and we extend one side, forming an exterior angle. The exterior angle is equal to the sum of the two angles inside the triangle that are not adjacent to the exterior angle. So in this case, the exterior angle is \(71^{\circ}\), one non - adjacent angle is \(49^{\circ}\), and the other non - adjacent angle is \(x\). So \(71=49 + x\)? No, that would give \(x = 22\), but that seems too small. Wait, no, I think I mixed up. Wait, actually, the angle adjacent to the exterior angle and the exterior angle form a linear pair (sum to \(180^{\circ}\)), but no, the exterior angle theorem is: exterior angle = sum of two remote interior angles. Wait, let's draw the triangle mentally. Let's say we have a triangle with angles \(x\), \(49^{\circ}\), and another angle. When we extend one side, we get an exterior angle of \(71^{\circ}\). The exterior angle is equal to the sum of the two angles that are not adjacent to it. So if the exterior angle is \(71^{\circ}\), and one of the non - adjacent angles is \(49^{\circ}\), and the other is \(x\), then \(71=x + 49\)? No, that would mean \(x=22\), but that can't be. Wait, no, maybe the exterior angle is equal to the sum of the two angles, but I think I got the angles wrong. Wait, no, let's use the fact that in a triangle, the sum of an interior angle and its adjacent exterior angle is equal to \(180^{\circ}\) only if they are supplementary, but no, the exterior angle is formed by extending a side, so the angle between the extended side and the other side is the exterior angle. Wait, maybe the correct formula is that the exterior angle is equal to the sum of the two remote interior angles. So if the exterior angle is \(71^{\circ}\), and one remote interior angle is \(49^{\circ}\), then the other remote interior angle \(x\) is such that \(71 = 49+x\)? No, that would be \(x = 22\), but let's check with the sum of angles in a triangle. Wait, the sum of angles in a triangle is \(180^{\circ}\). The angle adjacent to the exterior angle is \(180 - 71=109^{\circ}\), then \(x+49 + 109=180\), so \(x=180-(49 + 109)=180 - 158 = 22\). Wait, that's the same result. So \(x = 22\).

Step1: Apply the exterior angle theorem

The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. Let the exterior angle be \(E = 71^{\circ}\), one non - adjacent interior angle be \(A=49^{\circ}\), and the other non - adjacent interior angle be \(x\). Then by the exterior angle theorem:
\[E=A + x\]
Substitute \(E = 71^{\circ}\) and \(A = 49^{\circ}\) into the formula:
\[71=49 + x\]

Step2: Solve for \(x\)

Subtract \(49\) from both sides of the equation:
\[x=71 - 49\]
\[x = 22\]

Answer:

\(x = 22\)