Let's solve the problems step by step.

Problem 9:

We are given:

• $m \angle LKM = 7x - 5$
• $m \angle NKM = 3x + 9$

We know that $\overrightarrow{KN}$ bisects $\angle LKM$, which means: $m \angle LKN = m \angle NKM$

Since $m \angle LKM$ is the entire angle and $\overrightarrow{KN}$ bisects it, we have: $m \angle LKN = m \angle NKM = \frac{m \angle LKM}{2}$

Substituting the values: $3x + 9 = \frac{7x - 5}{2}$

Now, solve for $x$: $2(3x + 9) = 7x - 5$ $6x + 18 = 7x - 5$ $18 + 5 = 7x - 6x$ $x = 23$

Now, substitute $x = 23$ back into the expression for $m \angle LKM$: $m \angle LKM = 7(23) - 5 = 161 - 5 = 156^\circ$

Answer for Problem 9:

$m \angle LKM = 156^\circ$

Problem 10:

We are given:

• $m \angle NKL = 7x - 9$
• $m \angle JKM = x + 3$

Since $\overrightarrow{KJ}$ and $\overrightarrow{KL}$ are opposite rays, $\angle NKL + \angle JKM = 180^\circ$.

Therefore: $(7x - 9) + (x + 3) = 180$ $8x - 6 = 180$ $8x = 186$ $x = 23.25$

Now, substitute $x = 23.25$ back into the expression for $m \angle JKN$ (which is $m \angle JKM$ since the ray $\overrightarrow{KN}$ is the bisector): $m \angle JKN = 23.25 + 3 = 26.25^\circ$

Answer for Problem 10:

$m \angle JKN = 26.25^\circ$

Would you like more details on these steps or do you have any questions?

Here are 5 related questions you might be interested in:

1. How do you determine if two rays are opposite?
2. What is the significance of an angle bisector in geometry?
3. How do you find the value of an unknown angle in a triangle?
4. Can you explain why the sum of opposite rays' angles is always 180°?
5. How would you solve if the given angles involved different variables?

Tip: When dealing with equations involving angle bisectors, always set up your equations based on the property that the bisector divides the angle into two equal parts.