Question

given ( m parallel n ), find the value of ( x ).

(the image shows two parallel lines ( m ) and ( n ), cut by a transversal ( t ). the angle on line ( m ) is ( (10x - 1)^circ ), and the angle on line ( n ) is ( (x + 16)^circ ).)

answer attempt 1 out of 2
( x = ) input box submit answer

Explanation:

Step1: Identify Angle Relationship

Since \( m \parallel n \) and the transversal \( t \) intersects them, the angles \( (10x - 1)^\circ \) and \( (x + 16)^\circ \) are alternate interior angles. Alternate interior angles are equal when lines are parallel. So, \( 10x - 1 = x + 16 \).

Step2: Solve for \( x \)

Subtract \( x \) from both sides: \( 10x - x - 1 = 16 \) → \( 9x - 1 = 16 \).
Add 1 to both sides: \( 9x = 16 + 1 \) → \( 9x = 17 \)? Wait, no—wait, alternate interior angles? Wait, no, maybe they are same - side? Wait, no, looking at the diagram, actually, the angles \( (10x - 1)^\circ \) and \( (x + 16)^\circ \) are same - side interior angles? No, wait, no—wait, if \( m\parallel n \), and the transversal is crossing them, the angles \( (10x - 1) \) and \( (x + 16) \): wait, maybe they are supplementary? Wait, no, let's re - examine. Wait, the angle \( (10x - 1) \) and \( (x + 16) \): if \( m\parallel n \), and the transversal is t, then the angle \( (10x - 1) \) and the angle adjacent to \( (x + 16) \) would be corresponding, but actually, looking at the positions, \( (10x - 1) \) and \( (x + 16) \) are same - side interior angles? No, wait, no—wait, maybe they are equal? Wait, no, I think I made a mistake. Wait, the angle \( (10x - 1) \) and \( (x + 16) \): let's see, if the lines are parallel, and the transversal cuts them, then alternate interior angles are equal. Wait, maybe the angle \( (10x - 1) \) and \( (x + 16) \) are alternate interior angles. So set them equal: \( 10x - 1 = x + 16 \). Then, subtract \( x \): \( 9x - 1 = 16 \). Add 1: \( 9x = 17 \)? That can't be. Wait, maybe they are supplementary. Let's check the diagram again. Oh! Wait, maybe the angle \( (10x - 1) \) and \( (x + 16) \) are same - side interior angles, so they should be supplementary. So \( (10x - 1)+(x + 16)=180 \). Let's try that. So \( 10x - 1+x + 16 = 180 \). Combine like terms: \( 11x + 15 = 180 \). Subtract 15: \( 11x = 165 \). Divide by 11: \( x = 15 \). Ah, that makes sense. I must have misidentified the angle relationship. So same - side interior angles are supplementary when lines are parallel. So the correct equation is \( (10x - 1)+(x + 16)=180 \).

Step3: Correct Equation and Solve

Combine like terms: \( 10x - 1+x + 16 = 180 \) → \( 11x+15 = 180 \).
Subtract 15 from both sides: \( 11x = 180 - 15 \) → \( 11x = 165 \).
Divide both sides by 11: \( x=\frac{165}{11}=15 \).

Answer:

\( x = 15 \)