Question

in the figure below, ( l parallel m ). find ( x ).
(there is a figure showing two parallel lines ( l ) and ( m ), with a triangle between them having angles ( 60^circ ), ( 65^circ ), and ( x^circ ) at the intersection with line ( m ).)
( x = ) input box

Explanation:

Step1: Find the third angle of the triangle

The sum of angles in a triangle is \(180^\circ\). So the third angle of the triangle (let's call it \(y\)) is \(180 - 60 - 65\).
\(y = 180 - 60 - 65 = 55^\circ\)

Step2: Use the property of parallel lines (alternate interior angles or supplementary angles? Wait, actually, since \(l \parallel m\), the angle \(x\) and the angle \(y\) we found? Wait, no, wait. Wait, the straight line \(m\) has a straight angle (180 degrees). Wait, no, the triangle's base is on line \(m\)? Wait, no, the two parallel lines \(l\) and \(m\), and the triangle is between them. Wait, actually, the angle adjacent to \(x\) and the triangle's angles: Wait, no, let's re-examine. Wait, the sum of angles on a straight line (line \(m\)) is \(180^\circ\). Wait, no, the triangle has angles 60°, 65°, and the third angle. Then, since \(l \parallel m\), the angle \(x\) and the third angle of the triangle are equal? Wait, no, maybe I made a mistake. Wait, no, actually, the three angles at the intersection with line \(m\) should add up to \(180^\circ\) (since it's a straight line). Wait, the triangle's angles: 60°, 65°, and the angle adjacent to \(x\). Wait, let's correct. The triangle has angles 60°, 65°, so the third angle (let's call it \(z\)) is \(180 - 60 - 65 = 55^\circ\). Then, since \(l \parallel m\), the angle \(x\) and angle \(z\) are equal? Wait, no, wait, maybe the angle \(x\) and the third angle of the triangle are alternate interior angles? Wait, no, actually, the straight line \(m\) has the angles: \(x\), 65°, and the angle corresponding to 60°? Wait, no, I think I messed up. Wait, let's start over.

Wait, the two parallel lines \(l\) and \(m\), and a transversal? Wait, no, the triangle is formed by a vertex on line \(m\) and two vertices on line \(l\). So the base of the triangle is on line \(l\), and the vertex is on line \(m\). So the angles at the vertex on line \(m\) are \(x\), 65°, and the angle equal to 60° (because of alternate interior angles, since \(l \parallel m\)). Wait, yes! Because \(l \parallel m\), the angle between the triangle's side and line \(l\) (60°) is equal to the angle between the same side and line \(m\) (alternate interior angles). So then, the sum of \(x\), 65°, and 60° should be 180° (since they are on a straight line, line \(m\)). So:

\(x + 65 + 60 = 180\)

Wait, that makes sense. So:

\(x = 180 - 65 - 60\)

\(x = 55\)

Wait, but earlier when I calculated the triangle's angle, I got 55, and that's the same as \(x\). So that's consistent. So the key is that the sum of angles on a straight line (line \(m\)) is 180°, so \(x + 65 + 60 = 180\), so \(x = 180 - 65 - 60 = 55\).

Answer:

\(x = 55\)