Question

given ( m parallel n ), find the value of ( x ).
image shows two parallel lines ( m ) (top) and ( n ) (bottom), cut by a transversal ( t ). the angle on line ( m ) is ( (6x - 27)^circ ), and the angle on line ( n ) is ( (8x - 3)^circ ).

Explanation:

Step1: Identify vertical angles

The angle adjacent to $(6x - 27)^\circ$ and $(8x - 3)^\circ$ are vertical angles, so they are equal. Thus, $(6x - 27)^\circ + (8x - 3)^\circ = 180^\circ$ (supplementary angles on a straight line).

Step2: Combine like terms

$6x - 27 + 8x - 3 = 180$
$14x - 30 = 180$

Step3: Solve for x

$14x = 180 + 30$
$14x = 210$
$x = \frac{210}{14} = 15$

Wait, correction: Actually, since $m \parallel n$ and $t$ is transversal, the consecutive interior angles are supplementary. So $(6x - 27) + (8x - 3) = 180$.
Step1: Sum to 180
$6x -27 +8x -3 =180$
Step2: Simplify
$14x -30=180$
Step3: Solve
$14x=210$ → $x=15$.

Final correction: The correct value is 15.

Step1: Supplementary angles sum to 180

$(6x-27)+(8x-3)=180$

Step2: Combine like terms

$14x - 30 = 180$

Step3: Solve for x

$14x=210$ → $x=15$

Answer:

12