Question

1. 9. 10. 11. 12. 13. find the value of x so that ℓ || m.

Explanation:

Step1: Identify angle - relationship

When $\ell\parallel m$, the corresponding angles or alternate - interior angles are equal. For example, in the first case where we have $(2x + 6)^{\circ}$ and $130^{\circ}$, if they are corresponding or alternate - interior angles, we can set up an equation.
Let's assume they are equal: $2x+6 = 130$.

Step2: Solve the equation for $x$

Subtract 6 from both sides of the equation:
$2x=130 - 6$
$2x = 124$
Divide both sides by 2:
$x=\frac{124}{2}=62$.

For the second case with $(3x + 10)^{\circ}$ and $(4x-10)^{\circ}$, if they are alternate - interior angles (assuming $\ell\parallel m$), we set up the equation $3x + 10=4x-10$.
Subtract $3x$ from both sides: $10=x - 10$.
Add 10 to both sides: $x = 20$.

For the third case with $(6x + 4)^{\circ}$ and $(8x-8)^{\circ}$, set up the equation $6x + 4=8x-8$ (assuming they are corresponding or alternate - interior angles when $\ell\parallel m$).
Subtract $6x$ from both sides: $4 = 2x-8$.
Add 8 to both sides: $12 = 2x$.
Divide by 2: $x = 6$.

For the fourth case with $(4x)^{\circ}$ and $(x + 6)^{\circ}$, if they are alternate - interior angles, $4x=x + 6$.
Subtract $x$ from both sides: $3x=6$.
Divide by 3: $x = 2$.

For the fifth case with $(7x + 5)^{\circ}$ and $(5x+19)^{\circ}$, set up the equation $7x + 5=5x+19$ (assuming they are corresponding or alternate - interior angles when $\ell\parallel m$).
Subtract $5x$ from both sides: $2x+5 = 19$.
Subtract 5 from both sides: $2x=14$.
Divide by 2: $x = 7$.

For the sixth case with $(3x + 10)^{\circ}$ and $(5x+18)^{\circ}$, set up the equation $3x + 10=5x+18$ (assuming they are corresponding or alternate - interior angles when $\ell\parallel m$).
Subtract $3x$ from both sides: $10=2x + 18$.
Subtract 18 from both sides: $- 8=2x$.
Divide by 2: $x=-4$.

For the seventh case, if the angles are related such that $(7x + 15)^{\circ}$ and $25^{\circ}$ are corresponding or alternate - interior angles, we set up the equation $7x+15 = 25$.
Subtract 15 from both sides: $7x=25 - 15$
$7x = 10$
$x=\frac{10}{7}$.

Answer:

The values of $x$ for each case are: 62, 20, 6, 2, 7, - 4, $\frac{10}{7}$ respectively.