Question

question
a side of the triangle below has been extended to form an exterior angle of 68°.
find the value of x.
(there is a triangle image with angles 47°, x°, and an exterior angle of 68°)
answer attempt 1 out of 3
x = input box
submit answer

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. But here we can also use the fact that the sum of an interior angle and its adjacent exterior angle is \(180^{\circ}\), and the sum of the interior angles of a triangle is \(180^{\circ}\). First, we know that in a triangle, the sum of angles is \(180^{\circ}\). Let's consider the triangle with angles \(47^{\circ}\), \(x^{\circ}\), and the angle adjacent to \(68^{\circ}\). The angle adjacent to \(68^{\circ}\) is \(180 - 68=112^{\circ}\)? No, wait. Wait, the exterior angle is \(68^{\circ}\), so the adjacent interior angle is \(180 - 68 = 112^{\circ}\)? No, that's wrong. Wait, actually, the exterior angle and the adjacent interior angle are supplementary. But also, in the triangle, the sum of the three interior angles is \(180^{\circ}\). So we have \(47 + x+(180 - 68)=180\)? No, that's not right. Wait, let's start over. The exterior angle is equal to the sum of the two non - adjacent interior angles. So the exterior angle of \(68^{\circ}\) is equal to the sum of the two non - adjacent interior angles, which are \(47^{\circ}\) and \(x^{\circ}\)? Wait, no. Wait, the exterior angle is formed by extending a side, so the two non - adjacent interior angles to the exterior angle are the other two angles of the triangle. Wait, maybe I mixed up. Let's look at the diagram. The triangle has an angle of \(47^{\circ}\), an angle of \(x^{\circ}\), and the angle adjacent to the \(68^{\circ}\) exterior angle. Let's call the angle adjacent to \(68^{\circ}\) as \(y\). Then \(y + 68=180\) (supplementary angles), so \(y = 180 - 68=112\). Then, in the triangle, \(47+x + y=180\). Substitute \(y = 112\) into it: \(47+x + 112=180\). Then \(x+159 = 180\), so \(x=180 - 159 = 21\)? Wait, no, that can't be. Wait, maybe the exterior angle is equal to the sum of the two non - adjacent interior angles. So the exterior angle \(68^{\circ}\) should be equal to \(47^{\circ}+x^{\circ}\). Wait, is that the case? Let's check. If the exterior angle is equal to the sum of the two non - adjacent interior angles, then \(68 = 47+x\), so \(x=68 - 47 = 21\). Wait, that makes sense. Because 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. So in this case, the exterior angle of \(68^{\circ}\) is equal to the sum of the two non - adjacent interior angles, which are \(47^{\circ}\) and \(x^{\circ}\). So \(68=47 + x\).

Step2: Solve for x

We have the equation from the exterior angle theorem: \(x+47 = 68\). To solve for \(x\), we subtract \(47\) from both sides of the equation. So \(x=68 - 47\). Calculating \(68-47 = 21\).

Answer:

\(21\)