Question

has been extended to form an exterior angle of 132°. find

Explanation:

Step1: Find the adjacent interior angle

The exterior angle and its adjacent interior angle are supplementary, so the adjacent interior angle to \(132^{\circ}\) is \(180 - 132 = 48^{\circ}\).

Step2: Use triangle angle sum

In a triangle, the sum of interior angles is \(180^{\circ}\). Let the unknown angle be \(x\), and we know one angle is \(17^{\circ}\) and another is \(48^{\circ}\). So \(x + 17 + 48 = 180\).

Step3: Solve for \(x\)

Simplify the equation: \(x + 65 = 180\). Subtract 65 from both sides: \(x = 180 - 65 = 115\)? Wait, no, wait. Wait, no, the adjacent interior angle is \(180 - 132 = 48\), then the triangle has angles \(17\), \(x\), and the third angle? Wait, no, maybe I messed up. Wait, the exterior angle is formed by extending a side, so the two non - adjacent interior angles sum to the exterior angle? Wait, no, the exterior angle theorem: the exterior angle is equal to the sum of the two non - adjacent interior angles. Oh! I made a mistake earlier. So the exterior angle \(132^{\circ}\) is equal to the sum of the two non - adjacent interior angles, which are \(17^{\circ}\) and \(x\)? Wait, no, looking at the diagram, the exterior angle is at a vertex, and the two non - adjacent angles: wait, the triangle has angles \(17^{\circ}\), \(x\), and the angle adjacent to the exterior angle. Wait, the exterior angle and its adjacent interior angle are supplementary, so adjacent interior angle is \(180 - 132 = 48^{\circ}\). Then in the triangle, the sum of angles is \(180\), so \(17 + x+ 48 = 180\)? Wait, no, that would be if the three angles are \(17\), \(x\), and \(48\). But maybe the exterior angle is equal to the sum of the two remote interior angles. So \(132 = 17 + x\)? Wait, that can't be, because \(132 - 17 = 115\), but then the adjacent angle would be \(65\), and \(17+115 + 65=197\), which is wrong. Wait, I think I misidentified the angles. Let's re - examine. The exterior angle is \(132^{\circ}\), so the adjacent interior angle is \(180 - 132 = 48^{\circ}\). Then the triangle has angles: \(17^{\circ}\), \(x\), and \(48^{\circ}\)? No, that can't be, because \(17 + 48=65\), and \(180 - 65 = 115\), but then the sum would be \(17+115 + 48 = 180\). Wait, but the exterior angle theorem says that the exterior angle is equal to the sum of the two non - adjacent interior angles. So if the exterior angle is \(132\), then the two non - adjacent interior angles are \(17\) and \(x\)? No, that would mean \(17 + x=132\), so \(x = 132 - 17 = 115\), but then the adjacent angle is \(180 - 132 = 48\), and \(17+115 + 48=180\), which works. Wait, I see, I confused the exterior angle theorem. The exterior angle is equal to the sum of the two non - adjacent interior angles. So in this case, the exterior angle \(132^{\circ}\) is equal to the sum of the two non - adjacent interior angles, which are \(17^{\circ}\) and \(x\)? No, wait, no. Let's look at the diagram again. The side is extended, so the exterior angle is at a vertex, and the two angles inside the triangle that are not adjacent to the exterior angle: one is \(17^{\circ}\), the other is \(x\). Wait, no, the adjacent interior angle is \(180 - 132 = 48\), so the triangle has angles \(17\), \(x\), and \(48\). Then \(17+x + 48=180\), so \(x=180-(17 + 48)=180 - 65 = 115\)? But that seems off. Wait, no, maybe the exterior angle is equal to the sum of the two remote interior angles. So \(132=17 + x\), so \(x = 132 - 17 = 115\). Wait, but then the adjacent angle is \(180 - 132 = 48\), and \(17+115+48 = 180\), which is correct. So the correct approach is using the exterior angle theorem: the exterior angle is equal to the sum of the two non - adjacent interior angles. So \(132=17 + x\), so \(x = 132 - 17 = 115\)? Wait, no, that can't be, because \(17+115 = 132\), which matches the exterior angle theorem. Yes, the exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So here, the exterior angle is \(132^{\circ}\), and the two non - adjacent interior angles are \(17^{\circ}\) and \(x\). So \(x+17 = 132\), so \(x = 132 - 17 = 115\)? Wait, but then the third angle (adjacent to exterior angle) is \(180 - 132 = 48\), and \(17+115 + 48=180\), which is correct. So the value of \(x\) is \(115\)? Wait, no, wait, maybe I had the diagram wrong. Let me re - check. If the exterior angle is \(132\), and one of the non - adjacent angles is \(17\), then the other non - adjacent angle is \(x\), so \(x + 17=132\), so \(x = 115\). Yes, that's correct.

Answer:

\(115\)