Question

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

Explanation:

Step1: Use alternate interior angles

Since \( l \parallel m \), the alternate interior angle to \( 33^\circ \) is equal, so the triangle has angles \( 33^\circ \), \( 80^\circ \), and the third angle (let's call it \( y \)) which is equal to the alternate interior angle of \( 33^\circ \), but actually, we can use the fact that the sum of angles in a triangle is \( 180^\circ \), and then use linear pair or supplementary angles with \( x \). Wait, actually, the angle adjacent to \( x \) in the triangle: first, find the third angle of the triangle. The triangle has angles \( 33^\circ \), \( 80^\circ \), so the third angle \( z = 180 - 33 - 80 = 67^\circ \)? Wait, no, wait. Wait, the line \( l \) and \( m \) are parallel, so the angle with \( 33^\circ \) and the angle at the bottom left of the triangle are alternate interior angles, so they are equal. Then, the triangle's angles: let's see, the angle at the top is \( 33^\circ \), the angle inside the triangle is \( 80^\circ \), and the angle at the bottom left (alternate interior) is \( 33^\circ \). Wait, no, maybe better to use the fact that the sum of angles on a straight line is \( 180^\circ \). Wait, the angle \( x \) and the angle adjacent to it (let's call it \( a \)) are supplementary? No, wait, the triangle's angles: the three angles of the triangle are \( 33^\circ \), \( 80^\circ \), and \( (180 - x)^\circ \)? Wait, no, let's re-examine.

Wait, the two parallel lines \( l \) and \( m \), with a transversal forming a triangle. The angle of \( 33^\circ \) is on line \( l \), and the alternate interior angle is at the bottom left of the triangle, so that angle is \( 33^\circ \). Then the triangle has angles \( 33^\circ \), \( 80^\circ \), and the angle at the bottom right (let's call it \( b \)). So \( 33 + 80 + b = 180 \), so \( b = 180 - 33 - 80 = 67^\circ \). Then, since \( l \parallel m \), the angle \( x \) and \( b \) are supplementary? Wait, no, \( x \) and \( b \) are adjacent to a straight line? Wait, no, the angle \( x \) and the angle \( b \) are same-side interior angles? Wait, no, actually, the angle \( x \) and the angle \( (180 - b) \) would be... Wait, no, let's think again.

Wait, the angle at the top of the triangle (on line \( l \)) is \( 33^\circ \), the angle inside the triangle is \( 80^\circ \), and the angle at the bottom left (between the lower parallel line and the left side of the triangle) is equal to \( 33^\circ \) because of alternate interior angles (since \( l \parallel m \)). Then the triangle's angles sum to \( 180^\circ \), so the angle at the bottom right of the triangle (between the lower parallel line and the right side of the triangle) is \( 180 - 33 - 80 = 67^\circ \). Then, since the lower parallel line is a straight line, the angle \( x \) and the angle \( 67^\circ \) are supplementary? Wait, no, \( x \) is adjacent to that \( 67^\circ \) angle? Wait, no, the right side of the triangle makes an angle \( x \) with line \( m \), and the angle inside the triangle at the bottom right is \( 180 - x \)? Wait, no, I think I messed up.

Wait, correct approach: When two parallel lines are cut by a transversal, alternate interior angles are equal. The angle of \( 33^\circ \) on line \( l \) has an alternate interior angle at the bottom left of the triangle, so that angle is \( 33^\circ \). The triangle has angles \( 33^\circ \), \( 80^\circ \), and the third angle (let's call it \( c \)) at the top right of the triangle (on line \( l \))? No, wait, the triangle is formed by two lines from the intersection point on line \( l \) down to line \( m \). So the angles at the intersection point on line \( l \) are \( 33^\circ \), \( 80^\circ \), and the angle between the right side of the triangle and line \( l \), let's call that \( d \). So \( 33 + 80 + d = 180 \) (since they are on a straight line \( l \)), so \( d = 180 - 33 - 80 = 67^\circ \). Then, since \( l \parallel m \), the alternate interior angle to \( d \) is \( x \), so \( x = d = 67^\circ \)? Wait, no, alternate interior angles: the angle \( d \) on line \( l \) and angle \( x \) on line \( m \) are alternate interior angles, so they are equal. So \( x = 67^\circ \)? Wait, no, wait, let's check again.

Wait, the sum of angles on a straight line (line \( l \)): the three angles at the intersection point are \( 33^\circ \), \( 80^\circ \), and \( d \), so \( 33 + 80 + d = 180 \), so \( d = 67^\circ \). Then, since \( l \parallel m \), the angle \( x \) and angle \( d \) are alternate interior angles, so \( x = d = 67^\circ \)? Wait, no, that can't be, because the triangle's angle at the bottom right would be \( x \), but maybe I got the diagram wrong. Wait, maybe the angle \( x \) is supplementary to the angle we found. Wait, no, let's use the triangle angle sum and linear pair.

Wait, the triangle has angles: one angle is \( 33^\circ \) (alternate interior), one angle is \( 80^\circ \), so the third angle (at the bottom right of the triangle) is \( 180 - 33 - 80 = 67^\circ \). Then, since the bottom line \( m \) is straight, the angle \( x \) and that \( 67^\circ \) angle are supplementary? Wait, no, \( x + 67 = 180 \)? No, that would be \( x = 113 \), which is wrong. Wait, I think I made a mistake in identifying the angles.

Wait, let's start over. The two parallel lines \( l \) and \( m \). The transversal is the left side of the triangle, making a \( 33^\circ \) angle with line \( l \). So the alternate interior angle on line \( m \) is also \( 33^\circ \) (the angle at the bottom left of the triangle). Then the triangle has angles: \( 33^\circ \) (bottom left), \( 80^\circ \) (top), and the angle at the bottom right (let's call it \( e \)). So \( 33 + 80 + e = 180 \), so \( e = 67^\circ \). Now, the angle \( x \) is adjacent to \( e \) on line \( m \), so they form a linear pair? Wait, no, \( x \) and \( e \) are on a straight line? Wait, no, the right side of the triangle makes an angle \( x \) with line \( m \), and the angle \( e \) is inside the triangle. So actually, \( x \) is equal to \( 180 - e \)? Wait, no, that would be \( x = 113 \), but that doesn't make sense. Wait, maybe the angle \( x \) is equal to the sum of \( 33^\circ \) and \( 80^\circ \)? Wait, no, that would be \( 113 \), but let's check.

Wait, another approach: The exterior angle theorem. The exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. So the angle \( x \) is an exterior angle to the triangle, with the two non-adjacent interior angles being \( 33^\circ \) and \( 80^\circ \). So \( x = 33 + 80 = 113^\circ \). Ah, that makes sense! Because the exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two remote (non-adjacent) interior angles. So in this case, the angle \( x \) is an exterior angle, so it's equal to \( 33^\circ + 80^\circ = 113^\circ \).

Yes, that's the correct approach. So using the exterior angle theorem: the exterior angle \( x \) is equal to the sum of the two non-adjacent interior angles, which are \( 33^\circ \) and \( 80^\circ \).

Step2: Calculate \( x \)

\( x = 33 + 80 \)
\( x = 113 \)

Answer:

\( 113 \)