Question

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

Explanation:

Step1: Apply 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. In \(\triangle PNO\), \(x\) is an exterior angle. The two non - adjacent interior angles are \(38^{\circ}\) and \(39^{\circ}\). So \(x=38 + 39\) is not correct. But the exterior angle \(x\) is greater than either of the non - adjacent interior angles.

Step2: Analyze each option

• For the option \(x>38\): Since \(x\) is an exterior angle of the triangle and one of the non - adjacent interior angles is \(38^{\circ}\), by the exterior angle theorem (an exterior angle of a triangle is greater than either of the non - adjacent interior angles), \(x>38\).
• For the option \(x < 39\): Since \(x\) is an exterior angle and \(39^{\circ}\) is a non - adjacent interior angle, \(x>39\) (exterior angle theorem), so this option is wrong.
• For the option \(x < 77\): The sum of the two non - adjacent interior angles \(38+39 = 77\). But \(x\) is an exterior angle. If we consider the triangle's angle sum property, let the third interior angle of \(\triangle PNO\) be \(y\). Then \(y=180-(38 + 39)=103\). And \(x = 180 - y\). But using the exterior angle theorem (a more straightforward approach here), \(x>39\) and \(x>38\). Also, if we assume some wrong application of sum - of - angles, this option is incorrect.
• For the option \(x>103\): Let the third interior angle of \(\triangle PNO\) be \(y\). By the angle sum property of a triangle \(y=180-(38 + 39)=103\). And \(x=180 - y\), so \(x = 77\) (using angle - sum property of a straight line \(x+y = 180\)), so this option is wrong.

Answer:

\(x>38\)