Question

given ( m parallel n ), find the value of ( x ).
there is a diagram with two parallel lines ( m ) and ( n ), and a transversal ( ell ). the angles formed with the transversal are ( (5x + 19)^circ ) and ( (6x + 5)^circ ).
answer
attempt 1 out of 2
( x = )

Explanation:

Step1: Identify angle relationship

Since \( m \parallel n \) and the transversal \( \ell \) intersects them, the two angles \( (5x + 19)^\circ \) and \( (6x + 5)^\circ \) are alternate exterior angles (or corresponding, but here they are equal as alternate exterior for parallel lines). So set them equal: \( 5x + 19 = 6x + 5 \).

Step2: Solve for \( x \)

Subtract \( 5x \) from both sides: \( 19 = x + 5 \). Then subtract 5: \( x = 19 - 5 = 14 \). Wait, no—wait, maybe they are same - side? Wait, no, looking at the diagram, actually, if \( m \parallel n \), and the angles are on the same side? Wait, no, maybe I made a mistake. Wait, alternate exterior angles are equal. Wait, let's re - check. Wait, the angles \( (5x + 19)^\circ \) and \( (6x + 5)^\circ \): if \( m \parallel n \), and the transversal cuts them, then if they are alternate exterior angles, they are equal. Wait, but when we solve \( 5x + 19 = 6x + 5 \), we get \( x = 14 \). Wait, but let's check again. Wait, maybe they are supplementary? Wait, no, the diagram: the two angles are on the same side of the transversal? Wait, no, the first angle is above line \( m \), the second is below line \( n \)? Wait, no, maybe I misread. Wait, actually, when two parallel lines are cut by a transversal, same - side interior angles are supplementary, but alternate exterior are equal. Wait, let's look at the positions. The angle \( (5x + 19)^\circ \) and \( (6x + 5)^\circ \): if \( m \parallel n \), and the transversal \( \ell \) crosses them, then the two angles are alternate exterior angles, so they should be equal. So \( 5x + 19 = 6x + 5 \). Then, subtract \( 5x \) from both sides: \( 19 = x + 5 \). Then subtract 5: \( x = 14 \). Wait, but let's check: \( 5(14)+19 = 70 + 19 = 89 \), \( 6(14)+5 = 84 + 5 = 89 \). Oh, so they are equal. So that works.

Answer:

\( x = 14 \)