Question

in the figure below, ( l parallel m ). find ( x ).

Explanation:

Step1: Find adjacent angle to 151°

Since a straight line is 180°, the adjacent angle (let's call it \( y \)) to 151° on line \( m \) is \( 180 - 151 = 29^\circ \).

Step2: Use triangle angle sum and parallel lines

In the triangle, we know one angle is 101°, another is 29° (from step1, alternate interior angles as \( l \parallel m \)). The sum of angles in a triangle is 180°, so the third angle (let's call it \( z \)) in the triangle is \( 180 - 101 - 29 = 50^\circ \). But \( x \) and \( z \) are supplementary? Wait, no, actually, since \( l \parallel m \), the angle \( x \) and the angle we just found (50°) – wait, no, let's re-examine. Wait, the angle adjacent to \( x \) in the triangle: wait, maybe better to use linear pair and triangle. Wait, the 151° angle's supplement is 29°, which is equal to the angle inside the triangle (alternate interior). Then the triangle has angles 101°, 29°, and the third angle. Then \( x \) is equal to 180° minus (101° + 29°)? No, wait, \( x \) and the angle in the triangle (let's say angle \( A \)): since \( l \parallel m \), the angle \( x \) and the angle adjacent to the triangle on line \( l \) – wait, maybe simpler: the angle at the bottom left (adjacent to 151°) is 29°, then in the triangle, angles are 101°, 29°, so the third angle is 50°, and since \( x \) and that third angle are supplementary? No, wait, no. Wait, the line \( l \) and \( m \) are parallel, so the angle \( x \) and the angle we found (50°) – wait, no, actually, the angle \( x \) is equal to 180° - (101° + 29°)? Wait, no, let's do it again. The angle next to 151° is 29° (180 - 151). Then in the triangle, angles are 101°, 29°, so the third angle is 50°. Then, since \( l \) and \( m \) are parallel, the angle \( x \) and the 50° angle: wait, no, the angle \( x \) is adjacent to the 101° angle and the 50° angle? Wait, maybe the correct approach is: the angle supplementary to 151° is 29°, which is an alternate interior angle, so inside the triangle, one angle is 29°, another is 101°, so the third angle is 180 - 101 - 29 = 50°, and then \( x \) is equal to 180 - 101 - 29? No, wait, \( x \) is equal to 180 - (101 + 29)? Wait, no, \( x \) and the third angle (50°) are supplementary? No, that can't be. Wait, maybe I made a mistake. Wait, the two parallel lines \( l \) and \( m \), the transversal is the left side. Wait, the angle at the bottom is 151°, so its supplement is 29°, which is equal to the angle inside the triangle (alternate interior). Then the triangle has angles: 101°, 29°, and \( x \)? No, that's not right. Wait, no, the angle \( x \) is on line \( l \), and the triangle has angles: one at the top (x), one at the middle (101°), and one at the bottom (29°). Wait, no, the sum of angles in a triangle is 180°, so \( x + 101 + 29 = 180 \)? Then \( x = 180 - 101 - 29 = 50 \)? Wait, no, that would mean \( x = 50 \), but let's check again. Wait, the angle adjacent to 151° is 29°, which is equal to the angle inside the triangle (alternate interior), so that angle is 29°. Then the triangle has angles: 101°, 29°, and \( x \)? Wait, no, the angle \( x \) is on the line \( l \), so the three angles at the vertex (where the two lines meet) are \( x \), 101°, and the angle equal to 29°? Wait, no, the vertex has three angles: \( x \), 101°, and the angle that's equal to 29° (alternate interior). So those three angles should add up to 180° (since they form a straight line on \( l \)). So \( x + 101 + 29 = 180 \), so \( x = 180 - 101 - 29 = 50 \). Wait, that makes sense. So \( x = 50 \).

Answer:

\( \boxed{50} \)