Question

given ( m parallel n ), find the value of x.
diagram: two parallel lines ( m ) (top) and ( n ) (bottom) cut by transversal ( t ). angle on ( m ): ( (2x - 2)^circ ); angle on ( n ): ( (2x - 30)^circ ).
answer attempt 1 out of 2
( x = ) input box submit answer

Explanation:

Step1: Identify Alternate Interior Angles

Since \( m \parallel n \), the angles \( (2x - 2)^\circ \) and \( (2x - 30)^\circ \) are alternate interior angles? Wait, no, actually, looking at the diagram, the vertical angle of \( (2x - 30)^\circ \) and \( (2x - 2)^\circ \) should be equal? Wait, no, maybe they are alternate interior angles. Wait, no, let's correct. Actually, when two parallel lines are cut by a transversal, alternate interior angles are equal. Wait, but maybe the angles are equal because they are alternate interior angles. Wait, no, let's re - examine. The angle \( (2x - 2)^\circ \) and the angle that is vertical to \( (2x - 30)^\circ \) are equal. Wait, no, the correct approach: since \( m\parallel n \), the alternate interior angles are equal. Wait, the angle \( (2x - 2)^\circ \) and the angle supplementary to \( (2x - 30)^\circ \)? No, wait, maybe the two angles \( (2x - 2)^\circ \) and \( (2x - 30)^\circ \) are equal? No, that can't be. Wait, no, I think I made a mistake. Wait, the correct property: when two parallel lines are cut by a transversal, alternate interior angles are equal. Let's assume that the angle \( (2x - 2)^\circ \) and the angle that is vertical to \( (2x - 30)^\circ \) are alternate interior angles. Wait, the vertical angle of \( (2x - 30)^\circ \) is also \( (2x - 30)^\circ \) (vertical angles are equal). Wait, no, maybe the two angles \( (2x - 2)^\circ \) and \( (2x - 30)^\circ \) are equal? No, that would give \( 2x-2 = 2x - 30\), which is impossible. So I must have misidentified the angles. Wait, maybe the angle \( (2x - 2)^\circ \) and the angle \( (2x - 30)^\circ \) are same - side interior angles? No, same - side interior angles are supplementary. Wait, let's start over.

Wait, the correct approach: when two parallel lines \( m\) and \( n\) are cut by a transversal \( t\), the alternate interior angles are equal. Let's look at the diagram again. The angle \( (2x - 2)^\circ \) and the angle that is adjacent to \( (2x - 30)^\circ \) (the vertical angle of \( (2x - 30)^\circ \)) are alternate interior angles. Wait, the vertical angle of \( (2x - 30)^\circ \) is equal to \( (2x - 30)^\circ \), and if \( m\parallel n \), then \( 2x-2=2x - 30\)? No, that's not possible. Wait, I think I messed up the angle labels. Wait, maybe the angle \( (2x - 2)^\circ \) and the angle \( (2x - 30)^\circ \) are corresponding angles? No, corresponding angles are equal. Wait, maybe the two angles are equal because they are alternate interior angles. Wait, no, let's check the equation. Wait, maybe the problem is that the two angles are equal, so \( 2x-2=2x - 30\) is wrong. Wait, maybe there is a typo, or I misread the angles. Wait, maybe the angle is \( (2x + 30)^\circ \) instead of \( (2x - 30)^\circ \)? No, the user provided \( (2x - 30)^\circ \). Wait, maybe the angles are supplementary? Let's try that. If they are same - side interior angles, then \( (2x - 2)+(2x - 30)=180\). Let's solve that.

Step2: Set up the equation

If the two angles are same - side interior angles (since \( m\parallel n \), same - side interior angles are supplementary), then:
\( (2x - 2)+(2x - 30)=180 \)
Combine like terms:
\( 2x+2x-2 - 30 = 180 \)
\( 4x-32 = 180 \)

Step3: Solve for x

Add 32 to both sides:
\( 4x=180 + 32 \)
\( 4x=212 \)
Divide both sides by 4:
\( x=\frac{212}{4}=53 \)

Wait, but let's check again. If the angles are alternate interior angles, then \( 2x - 2=2x - 30\), which is impossible. So they must be same - side interior angles. So the correct equation is \( (2x - 2)+(2x - 30)=180 \), and solving gives \( x =…

Answer:

\( x = 53 \)