Question

given ( m parallel n ), find the value of ( x ) and ( y ).
(there is a diagram with two parallel lines ( m ) and ( n ), and a transversal intersecting them. angles ( x^circ ), ( y^circ ), and ( 121^circ ) are marked. at the bottom, there are answer boxes for ( x = ) and ( y = ), with attempt 1 out of 2 text.)

Explanation:

Step1: Find x using supplementary angles

The angle of \(121^\circ\) and the angle adjacent to it (which is equal to \(x\) because of corresponding angles or vertical angles, but first, the supplementary angle to \(121^\circ\) on line \(n\) is \(180 - 121 = 59^\circ\)? Wait, no. Wait, since \(m \parallel n\), the angle \(x\) and the angle supplementary to \(121^\circ\)? Wait, no, let's look at the diagram. The angle \(121^\circ\) and the angle that is vertical or corresponding to \(y\) or \(x\). Wait, actually, the angle \(121^\circ\) and \(y\) are same - side? No, wait, the angle \(121^\circ\) and the angle that is equal to \(y\) (since \(m \parallel n\), alternate interior angles or corresponding angles). Wait, no, let's correct. The angle \(121^\circ\) and the angle adjacent to it on the straight line is \(180 - 121=59^\circ\)? No, wait, the angle \(x\) and the angle that is equal to the supplementary angle of \(121^\circ\)? Wait, no, let's think again.

Wait, the two lines \(m\) and \(n\) are parallel, cut by a transversal. The angle \(121^\circ\) and the angle \(y\): are they corresponding angles? Wait, no, the angle \(121^\circ\) and \(y\) - if we look at the transversal, the angle \(121^\circ\) and \(y\) are actually same - side? No, wait, the angle \(x\) and the angle that is supplementary to \(121^\circ\) (because \(m\parallel n\), corresponding angles). Wait, the angle \(121^\circ\) and the angle adjacent to it (on the straight line) is \(180 - 121 = 59^\circ\)? No, that's not right. Wait, the angle \(x\) and the angle \(121^\circ\): since \(m\parallel n\), the angle \(x\) and the angle that is supplementary to \(121^\circ\) (because they are same - side exterior or interior? Wait, no, let's use the fact that consecutive angles on a straight line sum to \(180^\circ\), and corresponding angles are equal when lines are parallel.

Wait, the angle \(121^\circ\) and the angle \(y\): if we consider the transversal, the angle \(121^\circ\) and \(y\) are actually equal? No, that can't be, because \(121 + 121>180\). Wait, no, the angle \(121^\circ\) and the angle that is vertical to the angle adjacent to \(x\). Wait, maybe I made a mistake. Let's start over.

The sum of angles on a straight line is \(180^\circ\). So the angle adjacent to \(121^\circ\) (on the line \(n\)) is \(180 - 121=59^\circ\). Now, since \(m\parallel n\), the angle \(x\) is equal to this \(59^\circ\) angle (corresponding angles). Then, the angle \(y\) and \(121^\circ\) are equal (corresponding angles) or supplementary? Wait, no, the angle \(y\) and \(x\) are supplementary? Wait, no, \(x\) and \(y\) are on a straight line? Wait, in the diagram, the two angles \(x\) and \(y\) are adjacent and form a straight line? Wait, no, the transversal cuts \(m\), forming angles \(x\) and \(y\) which are adjacent, so \(x + y=180^\circ\)? Wait, no, if \(x\) and \(y\) are adjacent angles on a straight line, then \(x + y = 180^\circ\). But also, since \(m\parallel n\), the angle \(y\) is equal to \(121^\circ\) (corresponding angles), and \(x\) is equal to \(180 - 121 = 59^\circ\) (since \(x + y=180^\circ\)).

Yes, that makes sense. So:

Step1: Find y

Since \(m\parallel n\), the angle \(y\) and the \(121^\circ\) angle are corresponding angles (or alternate interior angles, depending on the diagram), so \(y = 121^\circ\)? Wait, no, that can't be, because if \(x\) and \(y\) are on a straight line, \(x + y=180\). So if \(y = 121\), then \(x=180 - 121 = 59\). Let's verify.

The angle \(121^\circ\) and \(y\): since \(m\parallel n\), the angle \(y\) is equal to \(121^\circ\) (corresponding angles, because the transversal cuts the two parallel lines, so the angle \(y\) and \(121^\circ\) are in the same position relative to the parallel lines and the transversal). Then, since \(x\) and \(y\) are adjacent angles forming a linear pair (they are on a straight line), \(x + y=180^\circ\). So \(x=180 - y\). Since \(y = 121^\circ\), then \(x=180 - 121 = 59^\circ\).

Step1: Determine y

Because \(m\parallel n\), the angle \(y\) and the \(121^\circ\) angle are corresponding angles (or alternate interior angles), so \(y = 121^\circ\).

Step2: Determine x

Since \(x\) and \(y\) form a linear pair (they are adjacent angles on a straight line), \(x + y=180^\circ\). Substitute \(y = 121^\circ\) into the equation: \(x=180 - 121=59^\circ\).

Answer:

\(x = 59\), \(y = 121\)