Final answer:
To determine the values of x for which lines j and k are parallel, solve each set of equations given for m∠L₁ and m∠L₂ by setting them equal to each other and solving for x, which yields the solutions x = 5/7 for the first set and x = 25/3 for the second set.
Step-by-step explanation:
To determine the value of x for which lines j and k are parallel (denoted as j || k), we need to find the values of x that make the measures of m∠L₁ and m∠L₂ equal, since parallel lines have corresponding angles that are equal in measure when cut by a transversal line. We are given two sets of equations for m∠L₁ and m∠L₂:
- m∠L₁ = 7x + 14
- m∠L₂ = 21x + 4
- m∠L₁ = 4x - 5
- m∠L₂ = x + 20
We'll need to set the equations from each set equal to each other and solve for x. For the first set:
- Equating the two angles: 7x + 14 = 21x + 4
- Solving the equation: Subtracting 7x from both sides gives 14 = 14x + 4
- Simplifying further: Subtract 4 from both sides to get 10 = 14x
- Dividing both sides by 14: x = 10/14, which simplifies to x = 5/7
For the second set:
- Equating the two angles: 4x - 5 = x + 20
- Solving the equation: Subtract x from both sides to get 3x - 5 = 20
- Adding 5 to both sides: 3x = 25
- Dividing both sides by 3: x = 25/3
Once x is found, we can plug it back into either original equation to find the measures of m∠L₁ and m∠L₂.