Question

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

Explanation:

Step1: Identify angle relations

Since \( l \parallel m \), the alternate interior angles and linear pair properties apply. First, find the supplementary angle to \( 101^\circ \) and \( 50^\circ \) in the triangle - like formation. Wait, actually, the sum of angles around a point or in the parallel line - transversal setup. Let's consider the angles on the transversal. The angle adjacent to \( 101^\circ \) and \( 50^\circ \) and \( x^\circ \) should sum up appropriately. Wait, another approach: the sum of angles in a triangle? No, actually, since \( l \parallel m \), the angle corresponding to the sum of \( 50^\circ \) and \( (180 - 101)^\circ \)? Wait, no. Let's correct: the angle inside the "triangle" formed by the transversal and the two parallel lines. The angle adjacent to \( 101^\circ \) is \( 180 - 101=79^\circ \)? No, wait, the three angles \( 50^\circ \), \( 101^\circ \), and the angle adjacent to \( x \) should be related. Wait, actually, the sum of angles on a straight line is \( 180^\circ \). So, the angle that is \( 180-(50 + 101)=29^\circ \)? No, that's not right. Wait, no, since \( l \parallel m \), the alternate interior angle for \( x \) and the angle formed by \( 50^\circ \) and the supplementary angle of \( 101^\circ \). Wait, the supplementary angle of \( 101^\circ \) is \( 180 - 101 = 79^\circ \). Then, since \( l \parallel m \), the angle \( x \) and the angle \( 50^\circ+79^\circ \)? No, that's not. Wait, let's look at the figure again. The two parallel lines \( l \) and \( m \), with a transversal creating a triangle - like shape with angles \( 50^\circ \), \( 101^\circ \), and \( x^\circ \). Wait, actually, the sum of angles in a triangle is \( 180^\circ \), but here, because of parallel lines, the angle \( x \) is equal to \( 101 - 50 \)? No, that's not. Wait, I think I made a mistake. Let's start over.

The key is that when two parallel lines are cut by a transversal, alternate interior angles are equal. Also, the sum of angles around a point or on a straight line is \( 180^\circ \). The angle of \( 101^\circ \) and the angle adjacent to \( x \) and \( 50^\circ \): Wait, the angle that is \( 180 - 101=79^\circ \) is not correct. Wait, no, the angle inside the "triangle" (the area between the two parallel lines and the transversal) has angles \( 50^\circ \), \( (180 - 101)^\circ=79^\circ \), and \( x^\circ \)? No, that can't be. Wait, no, the correct approach is: the angle \( x \) is equal to \( 101 - 50 \)? Wait, no, let's calculate the angle that is supplementary to \( 101^\circ \): \( 180 - 101 = 79^\circ \). Then, since \( l \parallel m \), the angle \( x \) and the angle \( 50^\circ+79^\circ \)? No, that's \( 129^\circ \), which is wrong. Wait, I think I messed up the figure. Let's assume that the three angles \( 50^\circ \), \( 101^\circ \), and \( x^\circ \) are related such that \( x=101 - 50 = 51^\circ \)? No, that's not. Wait, no, the correct way: the sum of angles on a straight line is \( 180^\circ \). So, \( 50+101 + x=180 \)? No, \( 50 + 101=151 \), \( 180 - 151 = 29 \), that's not. Wait, no, that's not the case. Wait, maybe the angle \( 101^\circ \) is an exterior angle. Wait, the exterior angle theorem: the exterior angle is equal to the sum of the two non - adjacent interior angles. So, if \( 101^\circ \) is an exterior angle, then \( 101 = 50 + x \), so \( x=101 - 50 = 51^\circ \). Ah, that makes sense! Because in the triangle formed by the two parallel lines and the transversal, the angle of \( 101^\circ \) is an exterior angle, and the two non - adjacent interior angles are \( 50^\circ \) and \( x^\circ \). By the exterior angle theorem, exterior angle = sum of two non - adjacent interior angles. So, \( 101=50 + x \), so \( x = 101 - 50=51 \).

Step2: Solve for \( x \)

Using the exterior angle theorem of a triangle (or the property of parallel lines and transversals which is equivalent here), we have the equation:
\( 101=50 + x \)
Subtract \( 50 \) from both sides:
\( x=101 - 50 \)
\( x = 51 \)

Answer:

\( 51 \)