Question

which statement about the value of x is true?
○ x > 38
○ x < 39
○ x < 77
○ x > 103
(diagram: triangle pno with point m on line no extended, angle at p is 38°, angle at o is 39°, angle at n (exterior, between m and n) is x°)

Explanation:

Step1: Find the interior angle at N in triangle PNO

In triangle \( PNO \), we know two angles: \( \angle P = 38^\circ \) and \( \angle O = 39^\circ \). The sum of the interior angles of a triangle is \( 180^\circ \). Let the interior angle at \( N \) be \( y \). So, \( y + 38^\circ + 39^\circ = 180^\circ \). Solving for \( y \), we get \( y = 180^\circ - 38^\circ - 39^\circ = 103^\circ \).

Step2: Relate the interior angle at N to \( x \)

The angle \( x \) and the interior angle \( y \) at \( N \) are supplementary (they form a linear pair), so \( x + y = 180^\circ \). We know \( y = 103^\circ \), so \( x = 180^\circ - 103^\circ = 77^\circ \)? Wait, no, wait. Wait, actually, the exterior angle \( x \) is equal to the sum of the two non - adjacent interior angles of the triangle. By the exterior angle theorem, \( x = 38^\circ+ 39^\circ=77^\circ \)? No, wait, no. Wait, the exterior angle at \( N \) (which is \( x \)) is equal to the sum of the two remote interior angles of triangle \( PNO \). The two remote interior angles are \( \angle P = 38^\circ \) and \( \angle O = 39^\circ \). So by the exterior angle theorem, \( x=38^{\circ}+ 39^{\circ}=77^{\circ} \)? Wait, no, that's not right. Wait, the interior angle at \( N \) is \( 180 - 38 - 39=103^{\circ} \), and then \( x \) and that interior angle are supplementary, so \( x = 180 - 103 = 77^{\circ} \)? Wait, no, I think I messed up. Wait, 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 in this case, the exterior angle at \( N \) (angle \( x \)) is equal to \( \angle P+\angle O \). So \( x = 38^{\circ}+39^{\circ}=77^{\circ} \)? Wait, no, that can't be. Wait, let's re - examine the diagram. The line \( MN \) is a straight line, so \( \angle MNO \) is a straight angle (\( 180^{\circ} \)). In triangle \( PNO \), the angles are \( \angle P = 38^{\circ} \), \( \angle O = 39^{\circ} \), so the third angle (at \( N \), inside the triangle) is \( 180 - 38 - 39 = 103^{\circ} \). Then, since \( \angle x \) and that \( 103^{\circ} \) angle are supplementary (they add up to \( 180^{\circ} \) because they are on a straight line), \( x=180 - 103 = 77^{\circ} \). Wait, but now let's check the options. The options are \( x>38 \), \( x < 39 \), \( x < 77 \), \( x>103 \). Wait, if \( x = 77 \), then \( x>38 \) is true (since \( 77>38 \)), \( x < 39 \) is false, \( x < 77 \) is false, \( x>103 \) is false. Wait, maybe my application of the exterior angle theorem was wrong. Wait, no, the exterior angle theorem: the exterior angle is equal to the sum of the two remote interior angles. So the exterior angle at \( N \) (angle \( x \)) should be equal to \( \angle P+\angle O \). So \( x = 38 + 39=77 \). So \( x = 77 \), so \( x>38 \) (since \( 77>38 \)) is true. Let's verify:

1. Calculate the interior angle at \( N \) in \( \triangle PNO \): \( 180-(38 + 39)=180 - 77 = 103^{\circ} \).
2. Since \( \angle x \) and the \( 103^{\circ} \) angle are supplementary, \( x = 180 - 103=77^{\circ} \).
3. Now check the options:

• \( x>38 \): \( 77>38 \), true.
• \( x < 39 \): \( 77<39 \), false.
• \( x < 77 \): \( 77<77 \), false.
• \( x>103 \): \( 77>103 \), false.

Answer:

\( x>38 \) (the option " \( x>38 \)")