Question

which statement about the value of x is true?
○ x > 38
○ x < 39
○ x < 77
○ x > 103

Explanation:

Step1: Apply exterior angle theorem

In a triangle, the exterior angle is equal to the sum of the two non - adjacent interior angles. Here, \(x\) is an exterior angle of \(\triangle PNO\). The two non - adjacent interior angles are \(38^{\circ}\) and \(39^{\circ}\). So, \(x=38 + 39\).

Step2: Calculate the value of \(x\)

\(x=38+39 = 77\). But wait, actually, the exterior angle \(x\) is greater than either of the non - adjacent interior angles. The formula for the exterior angle \(x\) of \(\triangle PNO\) (where \(x\) is the exterior angle at \(N\)) is \(x=\angle NPO+\angle PON\) (by the exterior angle theorem: \(x=\angle P+(180-(180 - x))=\angle P+\angle O\)). Since \(x=\angle P+\angle O\) and \(\angle P = 38^{\circ},\angle O=39^{\circ}\), and also, more accurately, the exterior angle \(x\) of a triangle is greater than any of the non - adjacent interior angles. So \(x>38\) (because \(x = 38 + 39=77\) and \(77>38\)).

Answer:

\(x > 38\)