Question

directions: if ( l parallel m ), solve for ( x ) and ( y ). 9. diagram with angles ( (9x + 25)^circ ), ( (13x - 19)^circ ), ( (17y + 5)^circ )

Explanation:

Step1: Identify angle relationship

Since \( l \parallel m \), the consecutive interior angles are supplementary? Wait, no, looking at the angles: \( (9x + 25)^\circ \) and \( (13x - 19)^\circ \) – wait, maybe they are alternate interior angles? Wait, no, maybe same - side interior? Wait, no, let's check the diagram. Wait, actually, if \( l \parallel m \), and the transversal cuts them, maybe \( (9x + 25) \) and \( (13x - 19) \) are same - side interior angles? Wait, no, maybe vertical angles? Wait, no, let's re - examine. Wait, the problem has angles \( (9x + 25)^\circ \), \( (13x - 19)^\circ \), and \( (17y + 5)^\circ \). Wait, maybe \( (9x + 25) \) and \( (13x - 19) \) are supplementary? No, wait, if \( l \parallel m \), and the two angles \( (9x + 25) \) and \( (13x - 19) \) are same - side interior angles? Wait, no, maybe they are equal? Wait, no, let's think again. Wait, maybe the angle \( (9x + 25) \) and \( (13x - 19) \) are same - side interior angles, so they should be supplementary? Wait, no, maybe alternate interior angles. Wait, I think I made a mistake. Wait, actually, if \( l \parallel m \), and the transversal is a line that intersects both, then \( (9x + 25) \) and \( (13x - 19) \) are same - side interior angles? Wait, no, let's solve for \( x \) first. Wait, maybe the two angles \( (9x + 25) \) and \( (13x - 19) \) are supplementary? Wait, no, let's set them equal? Wait, no, let's check the sum. Wait, maybe I misread the angles. Wait, the problem says "If \( l\parallel m \), solve for \( x \) and \( y \)". Let's assume that \( (9x + 25) \) and \( (13x - 19) \) are same - side interior angles, so their sum is \( 180^\circ \)? Wait, no, maybe they are alternate interior angles, so they are equal. Wait, let's try that.

Set \( 9x + 25=13x - 19 \)

Step2: Solve for \( x \)

Subtract \( 9x \) from both sides: \( 25 = 4x-19 \)

Add 19 to both sides: \( 25 + 19=4x \)

\( 44 = 4x \)

Divide both sides by 4: \( x = 11 \)

Now, let's find the measure of the angle. Substitute \( x = 11 \) into \( 9x + 25 \): \( 9\times11+25=99 + 25 = 124^\circ \)

Now, the angle \( (17y + 5)^\circ \) and the angle we just found (\( 124^\circ \)) – are they supplementary? Because if \( l\parallel m \), and the transversal, maybe they are same - side interior angles. So \( (17y + 5)+124 = 180 \)

Step3: Solve for \( y \)

\( 17y+5 + 124=180 \)

\( 17y+129 = 180 \)

Subtract 129 from both sides: \( 17y=180 - 129=51 \)

Divide both sides by 17: \( y = 3 \)

Answer:

\( x = 11 \), \( y = 3 \)