Question

question 4 of 10 what is the measure of ∠rsl? (9x + 27)° (6x + 66)° a. 142° b. 140° c. 146° d. 144°

Explanation:

Step1: Set up equation

Since $\angle RSL$ and $(6x + 66)^{\circ}$ are vertical - angles, they are equal. Also, $\angle RSL$ and $(9x + 27)^{\circ}$ are supplementary (linear - pair of angles), so $9x+27 + 6x + 66=180$.
$$9x+6x+27 + 66=180$$
$$15x+93 = 180$$

Step2: Solve for x

Subtract 93 from both sides of the equation:
$$15x=180 - 93$$
$$15x=87$$
Divide both sides by 15:
$$x=\frac{87}{15}=5.8$$

Step3: Find the measure of $\angle RSL$

Substitute $x = 5.8$ into the expression for $\angle RSL$ (using the vertical - angle relationship, $\angle RSL=6x + 66$):
$$\angle RSL=6\times5.8+66$$
$$\angle RSL = 34.8+66$$
$$\angle RSL=100.8$$
There seems to be an error above. Let's use the linear - pair relationship directly to find $\angle RSL$.
We know that $\angle RSL=180-(9x + 27)$.
First, solve $9x+27+6x + 66 = 180$ for $x$.
Combining like terms: $15x+93 = 180$, then $15x=180 - 93=87$, so $x=\frac{87}{15}=5.8$.
$\angle RSL=180-(9\times5.8 + 27)$
$=180-(52.2+27)$
$=180 - 79.2$
$=100.8$ (wrong).
Let's assume the correct linear - pair relationship:
Since $\angle RSL$ and $(9x + 27)$ are supplementary.
$9x+27+\angle RSL = 180$. Also, $\angle RSL=(6x + 66)$ (vertical angles).
So $9x+27+6x + 66=180$.
$15x+93 = 180$.
$15x=87$.
$x=\frac{87}{15} = 5.8$.
Using $\angle RSL=6x + 66$:
$\angle RSL=6\times5.8+66=34.8+66 = 100.8$ (wrong).
Let's start over.
Since $\angle RSL$ and $(9x + 27)$ are a linear pair, $\angle RSL=180-(9x + 27)$.
And since $\angle RSL$ and $(6x + 66)$ are vertical angles, $9x+27=6x + 66$ (because vertical angles are equal).

Step1: Solve for x

Subtract $6x$ from both sides: $9x-6x+27=6x-6x + 66$.
$3x+27 = 66$.
Subtract 27 from both sides: $3x=66 - 27$.
$3x=39$.
$x = 13$.

Step2: Find the measure of $\angle RSL$

Substitute $x = 13$ into the expression for $\angle RSL$ (using $\angle RSL=6x + 66$).
$\angle RSL=6\times13+66$.
$\angle RSL=78+66$.
$\angle RSL = 144^{\circ}$.

Answer:

D. $144^{\circ}$