Question

find x.
36°
30°
100°
x°
x =

Explanation:

Step1: Find the third angle of the triangle

First, we know that the angle adjacent to \(100^\circ\) on a straight line is \(180 - 100 = 80^\circ\). Then, in the triangle, the sum of angles is \(180^\circ\). So the third angle of the triangle (let's call it \(y\)) is \(180 - 36 - 30 - 80\)? Wait, no, wait. Wait, the triangle has angles \(36^\circ\), \(30^\circ\), and the angle adjacent to \(100^\circ\) is supplementary, so \(180 - 100 = 80^\circ\)? Wait, no, let's re-examine. The triangle's angles: one angle is \(36^\circ\), another is \(30^\circ\), and the third angle (let's call it \(z\)) is supplementary to \(100^\circ\)? Wait, no, the line is straight, so the angle adjacent to \(100^\circ\) is \(180 - 100 = 80^\circ\). Then, in the triangle, the sum of angles is \(180^\circ\), so \(36 + 30 + z = 180\)? Wait, no, that's not right. Wait, actually, the exterior angle or the vertical angles. Wait, maybe using the exterior angle theorem or vertical angles. Wait, let's find the angle at the vertex of the triangle. The angle adjacent to \(100^\circ\) is \(80^\circ\) (since they are supplementary). Then, the sum of angles in the triangle: \(36^\circ + 30^\circ + 80^\circ + \text{another angle}\)? No, wait, maybe the triangle has angles \(36^\circ\), \(30^\circ\), and the angle opposite to the angle we need. Wait, maybe using the exterior angle or the fact that vertical angles are equal. Wait, let's find the angle that is vertical to \(x\). First, let's find the angle at the top of the triangle. Wait, the triangle has angles: one is \(36^\circ\), another is \(30^\circ\), and the third angle (let's call it \(a\)) is such that \(a + 36 + 30 = 180 - (180 - 100)\)? No, this is confusing. Wait, let's do it step by step.

First, the angle adjacent to \(100^\circ\) is \(180 - 100 = 80^\circ\) (supplementary angles). Then, in the triangle, the sum of angles is \(180^\circ\), so the third angle of the triangle (let's call it \(b\)) is \(180 - 36 - 30 - 80\)? No, that can't be. Wait, no, the triangle has three angles: \(36^\circ\), \(30^\circ\), and the angle that is supplementary to \(100^\circ\)? Wait, no, the triangle's angles: one angle is \(36^\circ\), another is \(30^\circ\), and the third angle is \(180 - 36 - 30 - (180 - 100)\)? No, I'm overcomplicating. Let's use the exterior angle theorem or vertical angles.

Wait, the angle at the vertex of the triangle: the sum of \(36^\circ\) and \(30^\circ\) is \(66^\circ\). Then, the angle adjacent to \(100^\circ\) is \(80^\circ\) (supplementary). Then, the angle inside the triangle is \(180 - 36 - 30 - 80\)? No, that's \(34^\circ\)? Wait, no, \(36 + 30 + 80 = 146\), so \(180 - 146 = 34\). Then, the angle \(x\) is equal to that angle because they are vertical angles? Wait, no, maybe the angle we found is \(34^\circ\), but let's check again.

Wait, let's start over. The straight line has \(100^\circ\), so the adjacent angle is \(180 - 100 = 80^\circ\) (linear pair). Then, in the triangle, the angles are \(36^\circ\), \(30^\circ\), and \(80^\circ\)? No, that can't be, because \(36 + 30 + 80 = 146\), which is less than \(180\). Wait, no, the triangle has three angles: one is \(36^\circ\), another is \(30^\circ\), and the third angle is \(180 - 36 - 30 = 114^\circ\)? No, that's not right. Wait, maybe the angle of \(100^\circ\) is an exterior angle. Wait, the exterior angle theorem: the exterior angle is equal to the sum of the two non-adjacent interior angles. So, the exterior angle here is \(100^\circ\), so the sum of the two non-adjacent interior angles is \(100^\circ\). So, \(36^\circ + 30^\circ + x = 100^\circ\)? Wait, no, that would be \(x = 100 - 36 - 30 = 34\). Ah, that makes sense. So, using the exterior angle theorem, the exterior angle (\(100^\circ\)) is equal to the sum of the two non-adjacent interior angles (\(36^\circ\) and \(30^\circ\)) plus \(x\)? No, wait, maybe \(x\) is one of the non-adjacent angles. Wait, let's see: the angle \(x\) and the angles \(36^\circ\) and \(30^\circ\) are related to the exterior angle of \(100^\circ\). So, \(x + 36 + 30 = 100\), so \(x = 100 - 36 - 30 = 34\). Yes, that works.

Step2: Calculate \(x\)

So, \(x = 100 - 36 - 30\)
\(x = 34\)

Answer:

\(x = 34\)