Question

a side of the triangle below has been extended to form an exterior angle of 70°. find the value of x.
answer attempt 3 out of 6
x = 20%
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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 an exterior angle and its adjacent interior angle are supplementary (they add up to \(180^{\circ}\))? Wait, no, in this case, we can also use the fact that the sum of angles in a triangle is \(180^{\circ}\), and the exterior angle and the adjacent interior angle are supplementary. Wait, actually, let's look at the triangle. We have a triangle with one angle \(50^{\circ}\), another angle \(x\), and the exterior angle is \(70^{\circ}\). Wait, the exterior angle and the adjacent interior angle (let's call it \(y\)) satisfy \(y + 70^{\circ}=180^{\circ}\), so \(y = 180 - 70=110^{\circ}\)? No, that's not right. Wait, no, the exterior angle theorem states that the exterior angle is equal to the sum of the two remote interior angles. Wait, maybe I misread the diagram. Wait, the triangle has an angle of \(50^{\circ}\), an angle of \(x\), and the exterior angle is \(70^{\circ}\) which is adjacent to the third angle. Wait, actually, the sum of angles in a triangle is \(180^{\circ}\), and the exterior angle and its adjacent interior angle are supplementary. Wait, let's correct. The exterior angle is formed by extending a side, so the exterior angle and the adjacent interior angle are supplementary (sum to \(180^{\circ}\)), but also, the sum of the interior angles of a triangle is \(180^{\circ}\). Wait, in the triangle, we have angles: \(50^{\circ}\), \(x\), and the angle adjacent to the \(70^{\circ}\) exterior angle. Let's call the angle adjacent to the exterior angle \(z\). Then \(z+70^{\circ}=180^{\circ}\), so \(z = 110^{\circ}\)? No, that can't be, because the sum of angles in a triangle would be \(50 + x+z=180\), but if \(z = 110\), then \(50 + x+110=180\), so \(x=20\). Wait, that's the correct approach. Wait, the exterior angle is \(70^{\circ}\), so the adjacent interior angle is \(180 - 70=110\)? No, no, wait, maybe the exterior angle is equal to the sum of the two non - adjacent interior angles. Wait, no, 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. Wait, in this case, if the exterior angle is \(70^{\circ}\), and one of the non - adjacent interior angles is \(50^{\circ}\), and the other is \(x\), then \(70=50 + x\)? Wait, that would give \(x = 20\). Oh, right! The exterior angle theorem says that the exterior angle is equal to the sum of the two remote (non - adjacent) interior angles. So if the exterior angle is \(70^{\circ}\), and one remote interior angle is \(50^{\circ}\), and the other is \(x\), then \(70=50 + x\).

Step2: Solve for \(x\)

From the equation \(70 = 50+x\), we subtract \(50\) from both sides: \(x=70 - 50\)
\(x = 20\)

Answer:

\(x = 20\)