Question

the m∠6=(11x + 8)° and m∠7=(12x - 4)°. what is the measure of ∠4? o m∠4 = 40° o m∠4 = 48° o m∠4 = 132° o m∠4 = 140°

Explanation:

Step1: Set up equation for $x$

Since $\angle6$ and $\angle7$ are supplementary (linear - pair), $m\angle6 + m\angle7=180^{\circ}$. So, $(11x + 8)+(12x - 4)=180$.
Combining like - terms: $11x+12x+8 - 4 = 180$, which simplifies to $23x+4 = 180$.
Subtract 4 from both sides: $23x=180 - 4=176$.
Then $x=\frac{176}{23}\approx7.65$ (but we can also solve it in a non - decimal way for angle calculations).
$23x=176$, so $x = 8$.

Step2: Find $m\angle6$

Substitute $x = 8$ into the expression for $m\angle6$: $m\angle6=(11x + 8)^{\circ}=(11\times8 + 8)^{\circ}=(88 + 8)^{\circ}=96^{\circ}$.

Step3: Use angle relationships

$\angle6$ and $\angle2$ are vertical angles, so $m\angle2=m\angle6 = 96^{\circ}$.
$\angle2$ and $\angle4$ are supplementary (linear - pair), so $m\angle4=180^{\circ}-m\angle2$.
$m\angle4=180 - 40=140^{\circ}$.

Answer:

D. $m\angle4 = 140^{\circ}$