Question

find the value of k.
for the following intersection, the measurements of the indicated angles are represented by the expressions (mangle s=(6x + 78)^{circ}) and (mangle t=(6x + 174)^{circ}). what are the numerical angle measurements of (angle s) and (angle t)?

Explanation:

Step1: Set up the angle - sum equation

Since the two angles $(-4k + 140)^{\circ}$ and $(-3k + 75)^{\circ}$ are supplementary (they form a straight - line and their sum is $180^{\circ}$), we have the equation $(-4k + 140)+(-3k + 75)=180$.

Step2: Combine like terms

Combining the $k$ terms and the constant terms, we get $(-4k-3k)+(140 + 75)=180$, which simplifies to $-7k+215 = 180$.

Step3: Isolate the variable term

Subtract 215 from both sides of the equation: $-7k=180 - 215$, so $-7k=-35$.

Step4: Solve for k

Divide both sides by - 7: $k=\frac{-35}{-7}=5$.

For the second part:

Step1: Set up the angle - sum equation for supplementary angles

Since $\angle s$ and $\angle t$ are supplementary, $m\angle s+m\angle t = 180^{\circ}$. So, $(6x + 78)+(6x + 174)=180$.

Step2: Combine like terms

Combine the $x$ terms and the constant terms: $(6x+6x)+(78 + 174)=180$, which gives $12x+252 = 180$.

Step3: Isolate the variable term

Subtract 252 from both sides: $12x=180 - 252=-72$.

Step4: Solve for x

Divide both sides by 12: $x=\frac{-72}{12}=-6$.

Step5: Find the measure of $\angle s$

Substitute $x = - 6$ into the expression for $m\angle s$: $m\angle s=6x + 78=6\times(-6)+78=-36 + 78 = 42^{\circ}$.

Step6: Find the measure of $\angle t$

Substitute $x=-6$ into the expression for $m\angle t$: $m\angle t=6x + 174=6\times(-6)+174=-36 + 174 = 138^{\circ}$.

Answer:

$k = 5$
$m\angle s=42^{\circ}$
$m\angle t=138^{\circ}$