Let's solve the problems in the image step by step.

Problem 17

If $K$ is the incenter of $\triangle EFG$:

• $K$ is the point where the angle bisectors meet.
• The angle measures at $\triangle EFG$ must add up to $180^\circ$.

We have the given angles:

1. $\angle KEJ = 18x - 23$
2. $\angle JEG = 7x - 1$
3. $\angle EFG = 47^\circ$.

Solution: The sum of all angles in $\triangle EFG$ is: $(18x - 23) + (7x - 1) + 47 = 180$

Simplify: $25x + 23 = 180$

Solve for $x$: $25x = 157 \quad \implies \quad x = 6.28 \, \text{(approximately)}.$

Now, find each angle measure:

1. $\angle KEJ = 18x - 23 = 18(6.28) - 23 = 113.04 - 23 = 90.04^\circ$.
2. $\angle JEG = 7x - 1 = 7(6.28) - 1 = 43.96^\circ$.
3. $\angle EFG = 47^\circ$.
4. $\angle FGE = 180 - (\angle KEJ + \angle JEG) = 180 - (90.04 + 43.96) = 46^\circ$.

Problem 18

If $P$ is the centroid of $\triangle JKL$:

• A centroid divides each median into a $2:1$ ratio.

Given:

1. $PN = 5$,
2. $LM = 33$,
3. $KP = 16$,
4. $NL = 13$.

Solution:

1. $PM$: Since $P$ divides $LM$ into a $2:1$ ratio: $PM = \frac{2}{3} \times 33 = 22$

2. $PL$: Since $P$ divides $KL$ into a $2:1$ ratio: $PL = \frac{2}{3} \times KP = \frac{2}{3} \times 16 = 10.67$

3. $PO$: Since $P$ divides $NL$ into a $2:1$ ratio: $PO = \frac{2}{3} \times 13 = 8.67$

4. $JP$: From $JP = PN + PO$: $JP = 5 + 8.67 = 13.67$

5. $JN$: Sum $JP + PN$: $JN = 13.67 + 5 = 18.67$

6. $KL$: Total segment $KL = 3 \times KP / 2$: $KL = 3 \times 16 / 2 = 24$

Final Answers:

Problem 17:

• $x = 6.28$
• $m\angle KEJ = 90.04^\circ$
• $m\angle EFG = 47^\circ$
• $m\angle FGE = 46^\circ$
• $m\angle KGJ =$ (can be calculated based on geometry, assuming symmetry).

Problem 18:

• $PM = 22$
• $PL = 10.67$
• $PO = 8.67$
• $JP = 13.67$
• $JN = 18.67$
• $KL = 24$

Do you need further clarification or detailed explanations for any part of this solution?

Here are 5 additional questions to expand your understanding:

1. How can the centroid formula be derived geometrically?
2. What are the properties of an incenter in a triangle?
3. Why does the sum of angles in a triangle always equal $180^\circ$?
4. How do you confirm the $2:1$ ratio property of the centroid?
5. How can angle bisectors help in solving triangle properties?

Tip: Always double-check your calculations, especially when working with angle measures and triangle properties!