Respuesta :
Answer:
To find the value of \( x \), we need to use the properties of parallel lines cut by transversals.
Since lines \( l \) and \( m \) are parallel, corresponding angles are congruent. Similarly, since lines \( m \) and \( n \) are parallel, corresponding angles are also congruent.
Let's denote the given angles:
1. Angle \( 5x - 15 \) and angle \( 3x + 5 \) are corresponding angles cut by transversal \( m \).
2. Angle \( 5x - 15 \) and angle \( 6x - 10 \) are corresponding angles cut by transversal \( n \).
Since corresponding angles are congruent, we can set up the following equations:
1. \( 5x - 15 = 3x + 5 \) (corresponding angles from lines \( l \) and \( m \))
2. \( 5x - 15 = 6x - 10 \) (corresponding angles from lines \( l \) and \( n \))
Now, let's solve each equation:
1. \( 5x - 15 = 3x + 5 \)
Subtract \( 3x \) from both sides:
\( 5x - 3x - 15 = 5 \)
Combine like terms:
\( 2x - 15 = 5 \)
Add \( 15 \) to both sides:
\( 2x = 20 \)
Divide both sides by \( 2 \):
\( x = 10 \)
2. \( 5x - 15 = 6x - 10 \)
Subtract \( 5x \) from both sides:
\( -15 = x - 10 \)
Add \( 10 \) to both sides:
\( -5 = x \)
Since we have two different values of \( x \), let's consider both and verify which one satisfies our conditions.
For \( x = 10 \):
Angle \( 5x - 15 = 5(10) - 15 = 35 \)
Angle \( 3x + 5 = 3(10) + 5 = 35 \)
For \( x = -5 \):
Angle \( 5x - 15 = 5(-5) - 15 = -40 \)
Angle \( 6x - 10 = 6(-5) - 10 = -40 \)
Both values of \( x \) result in corresponding angles being congruent, which confirms that lines \( l \), \( m \), and \( n \) are parallel.
Therefore, the value of \( x \) can be either \( 10 \) or \( -5 \).
Step-by-step explanation:
