Let's break down the problem and find the measure of ∠LNK step by step.

Given:

• NK is the bisector of ∠LNM, meaning it divides the angle ∠LNM into two equal parts: ∠LNK and ∠NKM.
• m∠LNK = 3x + 12
• m∠NKM = 5x − 20
• m∠LNM = 2x + 12
• m∠KNM = 40°
• The sum of all angles in a triangle is 180°.

Step 1: Set up the equation for the bisected angles.

Since NK bisects ∠LNM, we know: $m∠LNK = m∠NKM$

Therefore, we can set: $3x + 12 = 5x - 20$

Step 2: Solve for x.

Now, solve this equation: $3x + 12 = 5x - 20$ Subtract $3x$ from both sides: $12 = 2x - 20$ Add 20 to both sides: $32 = 2x$ Divide by 2: $x = 16$

Step 3: Substitute $x = 16$ into ∠LNK.

Now, substitute $x = 16$ into the equation for $m∠LNK$: $m∠LNK = 3x + 12 = 3(16) + 12 = 48 + 12 = 60°$

So, m∠LNK = 60°.

Step 4: Find m∠LNM.

Since NK is the bisector of ∠LNM, the full angle is twice ∠LNK: $m∠LNM = 2 \times m∠LNK = 2 \times 60° = 120°$

However, the problem says: $m∠LNM = 2x + 12 = 2(16) + 12 = 32 + 12 = 44°$

This suggests a possible mistake in the initial setup, where the bisected angles might not sum up to this measure in the diagram provided.

Do you need details or have any questions?

Follow-up questions:

1. What is the sum of the interior angles in any triangle?
2. How do we solve equations involving variables and angles?
3. How can we use bisectors to solve angle problems in geometry?
4. How would the problem change if we had different angle measures?
5. How do we check our solution for errors when solving for $x$?

Tip: When using bisectors in geometry, remember that the bisector divides an angle into two equal parts.