Question

in the picture, line a is parallel to line b. m∠3 = 12x - 23 and m∠7 = 7x + 7

Explanation:

Step1: Identify angle - relationship

Since line $a$ is parallel to line $b$, $\angle3$ and $\angle7$ are corresponding angles, so $m\angle3 = m\angle7$.
$12x - 23=7x + 7$

Step2: Solve for $x$

Subtract $7x$ from both sides: $12x-7x - 23=7x-7x + 7$, which simplifies to $5x-23 = 7$.
Then add 23 to both sides: $5x-23 + 23=7 + 23$, so $5x=30$.
Divide both sides by 5: $x=\frac{30}{5}=6$.

Step3: Find $m\angle3$

Substitute $x = 6$ into the expression for $m\angle3$: $m\angle3=12x - 23=12\times6-23=72 - 23 = 49^{\circ}$.

Step4: Find $m\angle6$

$\angle3$ and $\angle6$ are same - side interior angles, so $m\angle3+m\angle6 = 180^{\circ}$.
$m\angle6=180 - m\angle3=180 - 49=131^{\circ}$.

Answer:

$x = 6$
$\angle3 = 49^{\circ}$
$\angle6 = 131^{\circ}$