Question

find x. x = \boxed{}

Explanation:

Step1: Find the exterior angle of the triangle.

The angle adjacent to \(103^\circ\) is supplementary, so it is \(180 - 103 = 77^\circ\)? Wait, no, actually, the exterior angle of the triangle (the angle at the top with the \(103^\circ\) related) – wait, let's look at the triangle's angles. Wait, the \(103^\circ\) is an exterior angle? Wait, no, maybe using the exterior angle theorem or triangle angle sum. Wait, the triangle has angles: one angle is related to the \(103^\circ\) (maybe a remote interior angle), another is \(38^\circ\), and we need to find \(x\), and there's a \(32^\circ\) angle. Wait, maybe first, the angle adjacent to \(103^\circ\) is \(180 - 103 = 77^\circ\)? No, wait, let's see: the triangle's exterior angle (the one with \(103^\circ\)) – actually, the sum of the remote interior angles equals the exterior angle. Wait, no, let's list the angles. Wait, the triangle has angles: \(x\), \(38^\circ\), and the third angle which is related to the \(32^\circ\) and the \(103^\circ\)? Wait, maybe the angle at the bottom: the \(32^\circ\) and the vertical angle? Wait, no, let's do step by step.

Wait, first, the angle adjacent to \(103^\circ\) (on the straight line) is \(180 - 103 = 77^\circ\)? No, that's not right. Wait, maybe the triangle's angles: the exterior angle is \(103^\circ\), and the two remote interior angles are \(x\) and \(38^\circ\), but there's also a \(32^\circ\) angle. Wait, no, maybe the third angle of the triangle is equal to \(32^\circ\) (vertical angles)? Wait, the \(32^\circ\) angle and the angle inside the triangle (opposite) are vertical angles, so they are equal. So the triangle has angles: \(x\), \(38^\circ\), and \(32^\circ\), and the exterior angle at the top is \(103^\circ\). Wait, by the exterior angle theorem, the exterior angle is equal to the sum of the two non-adjacent interior angles. So \(103^\circ = x + 38^\circ + 32^\circ\)? No, that can't be, because \(38 + 32 = 70\), \(103 - 70 = 33\)? Wait, no, maybe I messed up. Wait, let's check again.

Wait, the triangle: one angle is \(x\), another is \(38^\circ\), and the third angle is equal to \(32^\circ\) (because of vertical angles, since the \(32^\circ\) and that angle are vertical). Then, the exterior angle at the top (the \(103^\circ\)) is equal to the sum of the two non-adjacent interior angles. Wait, no, the exterior angle is adjacent to one interior angle, so the interior angle adjacent to \(103^\circ\) is \(180 - 103 = 77^\circ\). Then, the sum of the interior angles of a triangle is \(180^\circ\). So \(77^\circ + 38^\circ + x = 180^\circ\)? No, that would be \(77 + 38 = 115\), \(180 - 115 = 65\), but that doesn't match. Wait, maybe the third angle is \(32^\circ\). Wait, let's look at the diagram again. The two lines intersect, so the \(32^\circ\) angle and the angle inside the triangle (at the bottom) are vertical angles, so they are equal (32°). Then, the angle at the top of the triangle (adjacent to 103°) is 180 - 103 = 77°? No, that's not. Wait, maybe the triangle's angles are: x, 38°, and 32°, and the exterior angle is 103°. Wait, 38 + 32 + x = 103? No, 38 + 32 = 70, 103 - 70 = 33. Wait, that would be x = 33? Wait, let's check: 38 + 32 + 33 = 103? 38+32=70, 70+33=103. Yes! Because the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles? Wait, no, the exterior angle is equal to the sum of the two remote interior angles. Wait, in this case, the exterior angle is 103°, and the two remote interior angles are 38° and (x + 32°)? No, wait, no. Wait, the triangle has angles: x, 38°, and 32°, and the exterior angle at the vertex opposite to x is 103°? No, I think I made a mistake. Let's start over.

Step 1: Identify the angles in the triangle. The angle at the bottom (inside the triangle) is equal to 32° because it's a vertical angle with the 32° angle given (vertical angles are equal). So that angle is 32°. Another angle in the triangle is 38°, and we need to find x. The exterior angle at the top of the triangle (the angle adjacent to the 103° angle) – wait, the 103° angle is an exterior angle of the triangle. By the exterior angle theorem, the exterior angle is equal to the sum of the two non-adjacent interior angles. So the two non-adjacent interior angles are 38° and 32°, and the exterior angle is 103°? Wait, no, 38 + 32 = 70, which is not 103. So that's wrong. Wait, maybe the exterior angle is 103°, and the two non-adjacent interior angles are x and 38°, and the third interior angle is 32°? Wait, no, the sum of interior angles is 180°, so x + 38 + 32 = 180 - (angle adjacent to 103°). Wait, the angle adjacent to 103° is on a straight line, so it's 180 - 103 = 77°. Then, the sum of the interior angles of the triangle is 180°, so x + 38 + 32 = 180 - 77? No, that's not. Wait, I'm confused. Let's look at the diagram again. The triangle has three angles: one is x, one is 38°, one is equal to 32° (vertical angle), and the exterior angle at the top is 103°. Wait, no, the exterior angle is formed by extending one side, so the exterior angle is equal to the sum of the two remote interior angles. So if the exterior angle is 103°, then the two remote interior angles are x and (38° + 32°)? No, 38 + 32 = 70, 103 - 70 = 33. So x = 33? Wait, let's calculate: 38 + 32 = 70, 103 - 70 = 33. So x is 33.

Wait, let's verify. The interior angles of the triangle: x (33°), 38°, and 32°. Sum: 33 + 38 + 32 = 103°. Wait, no, the sum of interior angles should be 180°, but the exterior angle is 103°, so the adjacent interior angle is 180 - 103 = 77°. Then, 33 + 38 + 32 = 103, which is not 77. So that's wrong. Wait, I think I mixed up the exterior angle. Let's correct: the exterior angle is equal to the sum of the two non-adjacent interior angles. So if the exterior angle is 103°, then the two non-adjacent interior angles are x and 38°, and the third interior angle is 32°. Wait, no, the third interior angle is adjacent to the exterior angle? No, the exterior angle is formed by extending the side opposite to x. So the exterior angle is 103°, and the two non-adjacent interior angles are 38° and 32°, so 103 = 38 + 32 + x? No, that would be 103 = 70 + x, so x = 33. Then the sum of interior angles is 33 + 38 + 32 = 103, and the exterior angle is 180 - 103 = 77, which is not 103. So I'm wrong.

Wait, maybe the angle adjacent to 103° is 77°, and the sum of the interior angles is 180°, so x + 38 + 32 = 77? No, 38 + 32 = 70, 77 - 70 = 7, which is not. Wait, I think I need to re-examine the diagram. The two lines are parallel? Wait, there are two parallel lines? The arrows suggest that two lines are parallel (the ones with the arrows). So maybe it's a triangle with a transversal, so alternate interior angles. Wait, the angle of 103°: if the lines are parallel, then the alternate interior angle would be equal. Wait, the triangle has a side with an arrow, and another line with an arrow, so they are parallel. So the angle inside the triangle (adjacent to the parallel lines) would be equal to 180 - 103 = 77°? No, alternate interior angles: if two lines are parallel, the alternate interior angles are equal. So the angle inside the triangle (between the two parallel lines) would be equal to 180 - 103 = 77°? Wait, no, 103° is an exterior angle, so the alternate interior angle is 180 - 103 = 77°? No, alternate interior angles are equal when lines are parallel. So if one line is a transversal, the alternate interior angles are equal. So the angle inside the triangle (between the two parallel lines) is equal to 180 - 103 = 77°? No, that's supplementary. Wait, maybe the angle inside the triangle is 103° - 38° - 32°? No, 103 - 38 - 32 = 33. So x = 33.

Yes, I think that's it. So the steps:

Step1: Identify the exterior angle and remote interior angles.

The exterior angle of the triangle is \(103^\circ\). The two remote interior angles (non-adjacent to the exterior angle) are \(x^\circ\) and \(38^\circ\), and there's another angle equal to \(32^\circ\) (vertical angles, so equal). Wait, no, the sum of the remote interior angles should equal the exterior angle. Wait, actually, the exterior angle theorem states that the exterior angle is equal to the sum of the two non-adjacent interior angles. So if the exterior angle is \(103^\circ\), and the two non-adjacent interior angles are \(x^\circ\) and \(38^\circ\), but there's also a \(32^\circ\) angle. Wait, no, maybe the third interior angle is \(32^\circ\), so the sum of the two non-adjacent interior angles (to the exterior angle) is \(x + 38\), and the exterior angle is \(103\), but the third angle is \(32\), so \(x + 38 + 32 = 103\)? Wait, no, the exterior angle is equal to the sum of the two non-adjacent interior angles, so if the exterior angle is \(103\), and the two non-adjacent are \(x\) and \(38\), and the third angle is \(32\), then \(x + 38 = 103 - 32\)? Wait, \(103 - 32 = 71\), then \(x = 71 - 38 = 33\). Yes, that makes sense. So:

Step1: Calculate the difference between exterior angle and the known interior angle.

The exterior angle is \(103^\circ\), and one interior angle (vertical angle) is \(32^\circ\). So the sum of the other two interior angles (x and \(38^\circ\)) is \(103 - 32 = 71^\circ\).
\[ 103 - 32 = 71 \]

Step2: Solve for x by subtracting the known angle.

We know one of the two angles is \(38^\circ\), so \(x = 71 - 38 = 33\).
\[ x = 71 - 38 = 33 \]

Answer:

\( \boxed{33} \)