Question

12 if (mangle knm=(8x - 5)^{circ}) and (mangle mnj=(4x - 19)^{circ}), find the measure of (angle knm).

Explanation:

Step1: Note vertical - angle relationship

Since $\angle KNM$ and $\angle MNJ$ are vertical angles, they are congruent. So, $m\angle KNM=m\angle MNJ$.
$8x - 5=4x-19$

Step2: Solve for $x$

Subtract $4x$ from both sides:
$8x-4x - 5=4x-4x - 19$
$4x-5=-19$
Add 5 to both sides:
$4x-5 + 5=-19 + 5$
$4x=-14$
Divide both sides by 4:
$x=\frac{-14}{4}=-\frac{7}{2}$

Step3: Find the measure of $\angle KNM$

Substitute $x =-\frac{7}{2}$ into the expression for $m\angle KNM$:
$m\angle KNM=8x - 5=8\times(-\frac{7}{2})-5$
$=-28 - 5=-33$. But angle - measure cannot be negative. There is a mistake above. Since $\angle KNM$ and $\angle MNJ$ are a linear - pair (not vertical angles as mis - assumed before), $m\angle KNM+m\angle MNJ = 180^{\circ}$.
So, $(8x - 5)+(4x-19)=180$

Step4: Simplify the equation

Combine like terms:
$8x+4x-5 - 19=180$
$12x-24 = 180$

Step5: Solve for $x$

Add 24 to both sides:
$12x-24 + 24=180 + 24$
$12x=204$
Divide both sides by 12:
$x = 17$

Step6: Find the measure of $\angle KNM$

Substitute $x = 17$ into the expression for $m\angle KNM$:
$m\angle KNM=8x - 5=8\times17-5$
$=136 - 5=131^{\circ}$

Answer:

$131^{\circ}$