Question

question 7 of 10
what is the value of n?
133°
n°
142°
a. 85°
b. 38°
c. 81°
d. 47°

Explanation:

Step1: Find adjacent supplementary angles

The angle adjacent to \(133^\circ\) is \(180^\circ - 133^\circ = 47^\circ\). The angle adjacent to \(142^\circ\) is \(180^\circ - 142^\circ = 38^\circ\).

Step2: Use triangle angle sum property

In a triangle, the sum of interior angles is \(180^\circ\). Let the third angle (adjacent to \(n\)) be \(x\), then \(47^\circ + 38^\circ + x = 180^\circ\), so \(x = 180^\circ - 47^\circ - 38^\circ = 95^\circ\). Then \(n = 180^\circ - 95^\circ = 85^\circ\)? Wait, no, wait. Wait, actually, the angle \(n\) and the angle we just found (let's re - examine the diagram). Wait, maybe the angles at the vertices: the exterior angles and the triangle's angles. Wait, another approach: the sum of exterior angles of a triangle? No, the sum of the exterior angles of any polygon is \(360^\circ\), but for a triangle, the sum of two exterior angles and the supplementary angle of the third angle? Wait, no, let's look again.

Wait, the three angles around the triangle's vertices (the exterior - like angles) and the triangle's interior angles. Wait, actually, the angle \(n\), the angle supplementary to \(133^\circ\) (which is \(47^\circ\)), and the angle supplementary to \(142^\circ\) (which is \(38^\circ\)) form a triangle? No, wait, the correct way: the sum of the exterior angles of a triangle (considering the non - adjacent exterior angles) and the angle \(n\). Wait, no, let's use the fact that in the diagram, the angle \(n\) and the two angles we found (\(47^\circ\) and \(38^\circ\)): Wait, maybe I made a mistake. Let's start over.

The angle adjacent to \(133^\circ\) is \(180 - 133=47^\circ\) (interior angle of the triangle). The angle adjacent to \(142^\circ\) is \(180 - 142 = 38^\circ\) (interior angle of the triangle). Then, the third interior angle of the triangle is \(180-(47 + 38)=95^\circ\). Then, the angle \(n\) and this \(95^\circ\) angle are supplementary? Wait, no, if \(n\) is adjacent to this \(95^\circ\) angle, then \(n = 180 - 95=85^\circ\)? But wait, let's check the answer options. Option A is \(85^\circ\). Wait, but let's verify with 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. Wait, maybe the angle \(n\) is an exterior angle? Wait, no, the diagram: there is a triangle with two exterior angles marked as \(133^\circ\) and \(142^\circ\), and we need to find \(n\).

Wait, the sum of the exterior angles of a triangle (the three exterior angles) is \(360^\circ\). But if we have two exterior angles \(133^\circ\) and \(142^\circ\), and the third exterior angle is \(180 - n\) (since \(n\) and the third exterior angle are supplementary). Then \(133+142+(180 - n)=360\). Let's solve this: \(133 + 142+180 - n=360\), \(455 - n = 360\), so \(n = 455 - 360 = 95\)? No, that's not matching. Wait, I must have misinterpreted the diagram.

Wait, maybe the angle \(n\) is an interior angle, and the other two angles are exterior angles. Wait, the exterior angle theorem: the exterior angle is equal to the sum of the two non - adjacent interior angles. Let's say the exterior angle is \(133^\circ\), then it should be equal to \(n +\) (the angle adjacent to \(142^\circ\)). The angle adjacent to \(142^\circ\) is \(180 - 142 = 38^\circ\). So \(133=n + 38\), then \(n = 133 - 38 = 95\)? No, not in options. Wait, the other exterior angle: \(142=n+(180 - 133)\), \(142=n + 47\), so \(n = 142 - 47 = 95\)? Still no. Wait, the options are \(85,38,81,47\). So my approach is wrong.

Wait, let's look at the triangle's angles. The three angles: one is \(180 - 133 = 47\), one is \(180 - 142 = 38\), and the third angle (let's call it \(m\)) in the triangle. Then \(47+38 + m=180\), so \(m = 95\). Then the angle \(n\) and \(m\) are supplementary? No, \(n + m=180\)? Then \(n = 85\), which is option A. Wait, that must be it. So the angle \(n\) and the angle \(m\) (the interior angle of the triangle) are supplementary because they form a linear pair. So \(n = 180 - 95 = 85^\circ\).

Answer:

A. \(85^\circ\)